The 34-year-old will be out of contract this season at the Stadium of Light, his 14 Premier League goals this season proving in vain as the Black Cats were relegated to the Championship.
Pulis hinted that the England forward is one of several individuals they are pursuing, but admitted that there would be "enormous" costs even though he will be available on a free transfer.
"When you say he's a free agent, effectively he's a free agent, but that still brings an enormous cost to whichever club takes Jermain," Pulis told the Birmingham Mail.
"We've got our fingers in the pie on quite a few players and we'll see how we go on those players. If Jermain's one of them, we'll be pushing on that one if we can."
Defoe, recently recalled to the England squad by Gareth Southgate, is also thought to be on the radar of Bournemouth.