Fletcher, 30, has seen his first-team opportunities limited at United this season under new manager Louis van Gaal, with just four starts in 11 Premier League appearances.
The Scotland international has six months left on his current contract and, with more arrivals expected at Old Trafford in the future, is not expected to earn a new deal.
Sky Sports reported earlier this week that both West Ham United and Valencia are keen on his signature, while West Brom also saw an initial loan bid knocked back last week.
The Baggies are still eager to make a deal happen, however, with Pulis telling the press ahead of the weekend’s FA Cup tie at Birmingham: "He is talking to three or four clubs. We showed an interest and he'll make a decision and we'll wait and see.”
A one-club man, Fletcher has been at Old Trafford since the age of 11, making 223 appearances since signing his first professional contract in 2001.
He was diagnosed with chronic bowel disease ulcerative colitis in 2011, forcing him spend most of the next two seasons on the sidelines, before returning to the first-team setup in December 2013.
The Scotland skipper would no doubt prove a solid addition to the West Brom squad, although latest reports from the likes of the Standard have claimed that a switch to West Ham is imminent.