The 21-year-old midfielder joined the Cottagers from the Dutch side in January, but a lack of first-team chances have seen him reconsider his future.
"If [I could play more] at Fulham that is fine, otherwise I have to look further," David told Voetbal International. "Both clubs have been in touch, but I will hear from my agent.
"FC Twente is also a nice club, and I know the coaches well. Besides that I have many friends and family in the region."
David made three appearances in all competitions for Fulham during the 2013-14 campaign.