WebCL is expected to work in a similar way to WebGL, but to instead harness the compute power of modern graphics processors. There are currently a few basic WebCL demos for those running Mac OS X with a modern NVIDIA GPU that supports the OpenCL 1.0+ specification. Samsung is largely behind the work on bringing WebCL to WebKit while Nokia has been working on a WebCL extension for Mozilla Firefox. Those interested in learning more about WebCL can visit the Khronos Group Wiki page.
When WebGL initially came to web browsers on Linux, there were problems. In particular, the AMD Catalyst driver wasn't up to par, but generally more broken was the state of Mesa and its OpenGL ES 2.0 support across the different hardware drivers. It's not even until Firefox 6.0 where more GPU Linux drivers should be white-listed. On the WebCL side, there will be even greater problems than there was with supporting WebGL on an open-source stack. There currently is no working OpenCL support for the open-source GPU drivers.
Being worked on this summer as part of Google's Summer of Code (GSoC) is OpenCL over Gallium3D, which has had some code developed for a few years known as "Clover", but it's not yet in a usable form. Recently the OpenCL-Gallium3D milestone was hit of building native OpenCL kernels, but Clover remains out of master and still has a lot of work ahead before it's ready for adoption by end-users. Even when OpenCL-over-Gallium3D is there, it won't do any good for those with the Intel classic driver or those just wishing to use OpenCL on the CPU; well, unless the OpenCL state tracker is made to work with Softpipe or (more ideally) LLVMpipe. At least though the modern NVIDIA and AMD drivers support OpenCL 1.0+ when paired with modern graphics processors.