diff --git a/fsspec/implementations/http.py b/fsspec/implementations/http.py
index 093fa29be..41a49cc3d 100644
--- a/fsspec/implementations/http.py
+++ b/fsspec/implementations/http.py
@@ -326,7 +326,7 @@ async def gen_chunks():
         async with meth(self.encode_url(rpath), data=gen_chunks(), **kw) as resp:
             self._raise_not_found_for_status(resp, rpath)
 
-    async def _exists(self, path, **kwargs):
+    async def _exists(self, path, strict=False, **kwargs):
         kw = self.kwargs.copy()
         kw.update(kwargs)
         try:
@@ -334,8 +334,14 @@ async def _exists(self, path, **kwargs):
             session = await self.set_session()
             r = await session.get(self.encode_url(path), **kw)
             async with r:
+                if strict:
+                    self._raise_not_found_for_status(r, path)
                 return r.status < 400
+        except FileNotFoundError:
+            return False
         except aiohttp.ClientError:
+            if strict:
+                raise
             return False
 
     async def _isfile(self, path, **kwargs):
diff --git a/fsspec/implementations/http_sync.py b/fsspec/implementations/http_sync.py
index 08799f20a..a67ea3ea5 100644
--- a/fsspec/implementations/http_sync.py
+++ b/fsspec/implementations/http_sync.py
@@ -463,14 +463,20 @@ def _process_limits(self, url, start, end):
                 end -= 1  # bytes range is inclusive
         return f"bytes={start}-{end}"
 
-    def exists(self, path, **kwargs):
+    def exists(self, path, strict=False, **kwargs):
         kw = self.kwargs.copy()
         kw.update(kwargs)
         try:
             logger.debug(path)
             r = self.session.get(self.encode_url(path), **kw)
+            if strict:
+                self._raise_not_found_for_status(r, path)
             return r.status_code < 400
+        except FileNotFoundError:
+            return False
         except Exception:
+            if strict:
+                raise
             return False
 
     def isfile(self, path, **kwargs):
diff --git a/fsspec/implementations/tests/test_http.py b/fsspec/implementations/tests/test_http.py
index d014d1155..856b51811 100644
--- a/fsspec/implementations/tests/test_http.py
+++ b/fsspec/implementations/tests/test_http.py
@@ -163,6 +163,14 @@ def test_exists(server):
         h.cat(server.address + "/notafile")
 
 
+def test_exists_strict(server):
+    h = fsspec.filesystem("http")
+    assert not h.exists(server.address + "/notafile", strict=True)
+    with pytest.raises(aiohttp.ClientResponseError) as e:
+        h.exists(server.address + "/unauthorized", strict=True)
+    assert e.value.status == 401
+
+
 def test_read(server):
     h = fsspec.filesystem("http")
     out = server.realfile
diff --git a/fsspec/implementations/tests/test_http_sync.py b/fsspec/implementations/tests/test_http_sync.py
index 330cf4d07..5b0efab41 100644
--- a/fsspec/implementations/tests/test_http_sync.py
+++ b/fsspec/implementations/tests/test_http_sync.py
@@ -6,7 +6,7 @@
 import pytest
 
 import fsspec.utils
-from fsspec.tests.conftest import data, reset_files, server, win  # noqa: F401
+from fsspec.tests.conftest import data, requests, reset_files, server, win  # noqa: F401
 
 
 @pytest.fixture()
@@ -147,6 +147,14 @@ def test_exists(server, sync):
         h.cat(server.address + "/notafile")
 
 
+def test_exists_strict(server, sync):
+    h = fsspec.filesystem("http")
+    assert not h.exists(server.address + "/notafile", strict=True)
+    with pytest.raises(requests.exceptions.HTTPError) as e:
+        h.exists(server.address + "/unauthorized", strict=True)
+    assert e.value.response.status_code == 401
+
+
 def test_read(server, sync):
     h = fsspec.filesystem("http")
     out = server.address + "/index/realfile"
diff --git a/fsspec/tests/conftest.py b/fsspec/tests/conftest.py
index ccf7bdccf..0a56521ed 100644
--- a/fsspec/tests/conftest.py
+++ b/fsspec/tests/conftest.py
@@ -54,6 +54,7 @@ class HTTPTestHandler(BaseHTTPRequestHandler):
         "/simple/file": data,
         "/simple/dir/": _make_listing("/simple/dir/file"),
         "/simple/dir/file": data,
+        "/unauthorized": AssertionError("shouldn't access"),
     }
     dynamic_files = {}
 
@@ -85,6 +86,8 @@ def do_GET(self):
         if "redirect" in self.headers and file_path != "/index/realfile":
             new_url = _make_realfile(baseurl)
             return self._respond(301, {"Location": new_url})
+        if file_path == "/unauthorized":
+            return self._respond(401)
         if file_data is None:
             return self._respond(404)