I'd try to swap the telephone cable from the laptop to the desktop and vice-versa (unless you're using the same one for both of course in which acquire/beg and or steal one from somewhere else to test).
I'd also consider removing the modem driver and re-installing it from scratch, assuming you have said driver available.
Have you got call-waiting on the line? (some modems will put the Internet connection on hold to allow you to answer the call).
Failing all that there is an outside chance that it may be caused by a virus/spyware/malware/trojan etc.. but you'd expect it to cause more hassle and on a more regular basis so I'd be inclined to say that's not an issue.
Is there any noise on the line - while connected pick up a normal handset - is there any discernable interference which the desktop modem might be more susceptible to?
Having failed this *free* options I'd say a replacement of the modem might be an idea if the issue warrants it (i.e. can you live with it otherwise).
My $2 worth anyway.
Charles