The 25-year-old Scotland international midfielder has fallen down the pecking order at the Hawthorns since Clarke's arrival in the summer.
He made his 12th appearance of the season in the 2-0 defeat at Manchester United last weekend but was omitted from the 18-man squad for the 2-1 home loss to Fulham on New Year's Day.
Dorrans has already held talks with Clarke over his future and could now be on his way out of the club during the transfer window.
Clarke said: "Graham came to see me a couple of weeks ago, not just because of what has been happening this season but the previous season. He feels as though he needs a fresh challenge.
"We had an amicable discussion and sort of came to the agreement that if something came along in January that was good for the club and that was good for Graham then we would listen to the possibility that Graham could go."
Dorrans joined West Brom from Partick Thistle for £100,000 in July 2008 and signed a long-term contract in July 2010 which tied him to the club until the summer of 2015.