diff --git a/Lib/importlib/_bootstrap.py b/Lib/importlib/_bootstrap.py
--- a/Lib/importlib/_bootstrap.py
+++ b/Lib/importlib/_bootstrap.py
@@ -21,6 +21,13 @@ work. One should use importlib as the pu
 
 CASE_INSENSITIVE_PLATFORMS = 'win', 'cygwin', 'darwin'
 
+
+def _relax_case():
+    """True if filenames must be checked case-insensitively."""
+    return (any(map(sys.platform.startswith, CASE_INSENSITIVE_PLATFORMS)) and
+            b'PYTHONCASEOK' in _os.environ)
+
+
 def _case_insensitive_ok(directory, check):
     """Check if the directory contains something matching 'check' exists in the
     directory.
@@ -760,29 +767,53 @@ class _FileFinder:
                                 for suffix in detail.suffixes)
         self.packages = packages
         self.modules = modules
-        self.path = path
+        # Base (directory) path
+        self.path = path or '.'
+        self._path_mtime = -1
+        self._path_cache = set()
 
     def find_module(self, fullname):
         """Try to find a loader for the specified module."""
         tail_module = fullname.rpartition('.')[2]
-        base_path = _path_join(self.path, tail_module)
-        if _path_isdir(base_path) and _case_ok(self.path, tail_module):
-            for suffix, loader in self.packages:
-                init_filename = '__init__' + suffix
-                full_path = _path_join(base_path, init_filename)
-                if (_path_isfile(full_path) and
-                        _case_ok(base_path, init_filename)):
-                    return loader(fullname, full_path)
-            else:
-                msg = "Not importing directory {}: missing __init__"
-                _warnings.warn(msg.format(base_path), ImportWarning)
+        if _relax_case:
+            tail_module = tail_module.lower()
+        try:
+            mtime = _os.stat(self.path).st_mtime
+        except OSError:
+            mtime = -1
+        if mtime != self._path_mtime:
+            self._fill_cache()
+            self._path_mtime = mtime
+        cache = self._path_cache
+        if tail_module in cache:
+            base_path = _path_join(self.path, tail_module)
+            if _path_isdir(base_path):
+                for suffix, loader in self.packages:
+                    init_filename = '__init__' + suffix
+                    full_path = _path_join(base_path, init_filename)
+                    if _path_isfile(full_path):
+                        return loader(fullname, full_path)
+                else:
+                    msg = "Not importing directory {}: missing __init__"
+                    _warnings.warn(msg.format(base_path), ImportWarning)
         for suffix, loader in self.modules:
             mod_filename = tail_module + suffix
-            full_path = _path_join(self.path, mod_filename)
-            if _path_isfile(full_path) and _case_ok(self.path, mod_filename):
-                return loader(fullname, full_path)
+            if mod_filename in cache:
+                full_path = _path_join(self.path, mod_filename)
+                if _path_isfile(full_path):
+                    return loader(fullname, full_path)
         return None
 
+    def _fill_cache(self):
+        """Fill the cache of potential modules and packages for this directory."""
+        path = self.path
+        l = _os.listdir(path)
+        if _relax_case:
+            self._path_cache = set(fn.lower() for fn in l)
+        else:
+            self._path_cache = set(l)
+
+
 class _SourceFinderDetails:
 
     loader = _SourceFileLoader
@@ -1093,6 +1124,8 @@ def _setup(sys_module, imp_module):
         raise ImportError('importlib requires posix or nt')
     setattr(self_module, '_os', os_module)
     setattr(self_module, 'path_sep', path_sep)
+    # Constants
+    setattr(self_module, '_relax_case', _relax_case())
 
     if sys_module.platform in CASE_INSENSITIVE_PLATFORMS:
         _case_ok = _case_insensitive_ok