Unfortunately, no. And I believe this is by design.
The issue is that the Windows clipboard does not have to store the data. It is effectively a clearing house where any application can list the data that was just copied (or cut) and offer it up to any application (including itself) to be pasted. The clipboard contains a list of formats in which the data is available, and if the application chooses to do so, the data itself in some of those formats.
The application that processes a Paste operation then has the option to choose the data format that best suits it. Some applications (e.g. Notepad) will only accept one particular format (such as plain text) and simply do nothing if that format is not available. Others pick a preferred format by default but provide a UI for choosing among alternatives that can be understood by that application.
In principle a program could be written that monitors the clipboard for new content and flattens any non-text content to just plain text. There is an event available to notify such a monitor of new content on the clipboard. However, I suspect that implementing such a thing would break a fair number of programs, and possibly in surprising ways.
The simple workaround in stock XP is to use Notepad to flatten the data when needed, or to use one of the many tools that improve the clipboard by implementing a stack, providing a view of its content, and so forth.
If you want to try building a tool yourself, might benefit from this question over at SO, and its related questions....