The Cottagers are said to have made an approach for the Bulgarian shot-stopper during the winter transfer window.
They saw their apparent efforts to do a deal knocked back as Twente would have been left with little time in which to find a replacement.
It remains to be seen whether Fulham will return in the summer, with Martin Jol having expressed a desire to bolster his keeping ranks.
He will, however, not be alone in chasing Mihaylov's signature should a second move be made for a man who took in three years on Liverpool's books before heading for Holland in 2010.
The 24-year-old keeper told De Telegraaf: "There are more clubs interested than just Fulham.
"However, at the moment I want to focus fully on my job at Twente."