diff --git a/Doc/library/tarfile.rst b/Doc/library/tarfile.rst
--- a/Doc/library/tarfile.rst
+++ b/Doc/library/tarfile.rst
@@ -77,6 +77,9 @@ Some facts and figures:
    is not suitable to open a certain (compressed) file for reading,
    :exc:`ReadError` is raised. Use *mode* ``'r'`` to avoid this.  If a
    compression method is not supported, :exc:`CompressionError` is raised.
+   As a :term:`LBYL` alternative to catching this exception, a program can
+   inspect :data:`tarfile.compression_formats` to find what formats are
+   supported.
 
    If *fileobj* is specified, it is used as an alternative to a :term:`file object`
    opened in binary mode for *name*. It is supposed to be at position 0.
@@ -207,6 +210,14 @@ The following variables are available on
    :func:`sys.getfilesystemencoding` otherwise.
 
 
+.. data:: compression_formats
+
+   A list containing all supported compression formats, sorted by strength; it
+   may contain ``'xz'``, ``'bz2'`` and ``'gz'``, in this order.
+
+   .. versionadded:: 3.3
+
+
 .. seealso::
 
    Module :mod:`zipfile`
diff --git a/Lib/tarfile.py b/Lib/tarfile.py
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -53,6 +53,21 @@ try:
 except ImportError:
     grp = pwd = None
 
+try:
+    import zlib
+except ImportError:
+    zlib = None
+
+try:
+    import lzma
+except ImportError:
+    lzma = None
+
+try:
+    import bz2
+except ImportError:
+    bz2 = None
+
 # os.symlink on Windows prior to 6.0 raises NotImplementedError
 symlink_exception = (AttributeError, NotImplementedError)
 try:
@@ -63,7 +78,8 @@ except NameError:
     pass
 
 # from tarfile import *
-__all__ = ["TarFile", "TarInfo", "is_tarfile", "TarError"]
+__all__ = ["TarFile", "TarInfo", "is_tarfile", "TarError",
+           "compression_formats"]
 
 from builtins import open as _open # Since 'open' is TarFile.open
 
@@ -413,11 +429,8 @@ class _Stream:
 
         try:
             if comptype == "gz":
-                try:
-                    import zlib
-                except ImportError:
+                if zlib is None:
                     raise CompressionError("zlib module is not available")
-                self.zlib = zlib
                 self.crc = zlib.crc32(b"")
                 if mode == "r":
                     self._init_read_gz()
@@ -426,9 +439,7 @@ class _Stream:
                     self._init_write_gz()
 
             elif comptype == "bz2":
-                try:
-                    import bz2
-                except ImportError:
+                if bz2 is None:
                     raise CompressionError("bz2 module is not available")
                 if mode == "r":
                     self.dbuf = b""
@@ -438,9 +449,7 @@ class _Stream:
                     self.cmp = bz2.BZ2Compressor()
 
             elif comptype == "xz":
-                try:
-                    import lzma
-                except ImportError:
+                if lzma is None:
                     raise CompressionError("lzma module is not available")
                 if mode == "r":
                     self.dbuf = b""
@@ -465,10 +474,10 @@ class _Stream:
     def _init_write_gz(self):
         """Initialize for writing with gzip compression.
         """
-        self.cmp = self.zlib.compressobj(9, self.zlib.DEFLATED,
-                                            -self.zlib.MAX_WBITS,
-                                            self.zlib.DEF_MEM_LEVEL,
-                                            0)
+        self.cmp = zlib.compressobj(9, zlib.DEFLATED,
+                                       -zlib.MAX_WBITS,
+                                       zlib.DEF_MEM_LEVEL,
+                                       0)
         timestamp = struct.pack("<L", int(time.time()))
         self.__write(b"\037\213\010\010" + timestamp + b"\002\377")
         if self.name.endswith(".gz"):
@@ -480,7 +489,7 @@ class _Stream:
         """Write string s to the stream.
         """
         if self.comptype == "gz":
-            self.crc = self.zlib.crc32(s, self.crc)
+            self.crc = zlib.crc32(s, self.crc)
         self.pos += len(s)
         if self.comptype != "tar":
             s = self.cmp.compress(s)
@@ -526,7 +535,7 @@ class _Stream:
     def _init_read_gz(self):
         """Initialize for reading a gzip compressed fileobj.
         """
-        self.cmp = self.zlib.decompressobj(-self.zlib.MAX_WBITS)
+        self.cmp = zlib.decompressobj(-zlib.MAX_WBITS)
         self.dbuf = b""
 
         # taken from gzip.GzipFile with some alterations
@@ -1790,9 +1799,7 @@ class TarFile(object):
         if len(mode) > 1 or mode not in "rw":
             raise ValueError("mode must be 'r' or 'w'.")
 
-        try:
-            import bz2
-        except ImportError:
+        if bz2 is None:
             raise CompressionError("bz2 module is not available")
 
         fileobj = bz2.BZ2File(filename=name if fileobj is None else None,
@@ -1814,9 +1821,7 @@ class TarFile(object):
         if mode not in ("r", "w"):
             raise ValueError("mode must be 'r' or 'w'")
 
-        try:
-            import lzma
-        except ImportError:
+        if lzma is None:
             raise CompressionError("lzma module is not available")
 
         fileobj = lzma.LZMAFile(filename=name if fileobj is None else None,
@@ -2559,9 +2564,9 @@ class TarIter:
         self.index += 1
         return tarinfo
 
-#--------------------
-# exported functions
-#--------------------
+#---------------------------------
+# exported functions and constants
+#---------------------------------
 def is_tarfile(name):
     """Return True if name points to a tar archive that we
        are able to handle, else return False.
@@ -2575,3 +2580,8 @@ def is_tarfile(name):
 
 bltn_open = open
 open = TarFile.open
+
+# supported compression formats, sorted by strength
+compression_formats = [name for (name, module) in
+                       [("xz", lzma), ("bz2", bz2), ("gz", zlib)]
+                       if module is not None]
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -1657,6 +1657,18 @@ class MiscTest(unittest.TestCase):
         self.assertRaises(ValueError, tarfile.itn, -0x10000000001, 6, tarfile.GNU_FORMAT)
         self.assertRaises(ValueError, tarfile.itn, 0x10000000000, 6, tarfile.GNU_FORMAT)
 
+    def test_compression_formats(self):
+        expected = []
+        # make sure the available formats are sorted by strength
+        if lzma is not None:
+            expected.append("xz")
+        if bz2 is not None:
+            expected.append("bz2")
+        if gzip is not None:
+            expected.append("gz")
+        self.assertEqual(tarfile.compression_formats, expected)
+        # test that if an item is deleted, nothing changes?
+
 
 class ContextManagerTest(unittest.TestCase):