Browne, out of the game since being sacked by the Hammers in November, revealed he has held talks with Potters chairman Steve Jefferis with a view to replacing Adie Arnold who quit last Saturday.
Jefferis confirmed on Wednesday that he had spoken to two prospective candidates, one of which was Browne, over the permanent position after coach Danny Saxby had agreed to stay on as interim manager “for the foreseeable future”.
Confirming his interest, Browne, who spent four years in charge of Hamworthy, told the Daily Echo: “I had an informal chat with the chairman this week but that is as far as it has gone at the moment.
“Having been out of the game for four months, my feet have started to get itchy on Saturdays and I would like to get back into the game.
"I have really enjoyed my time off with the family but have missed being involved with football and feel ready now.
“Verwood may well have other people they want to talk to so we will have to see what happens. The way we left it was that they would keep me posted.”
Former Cherries favourite John Sainty, who managed Chester City in the Football League, is also understood to be interested in the post.