TEAMtalk reported last week that Stoke have enquired about the possibility of signing Grant on loan.
The Potters secured a 4-0 home win over Reading on Saturday which relieved some of the pressure on manager Alex Neil and moved them up to 18th in the Championship table, seven points above the relegation zone.
But they are still hoping to boost their attacking options before the window closes at the end of January - and Grant is believed to be high on their wanted list.
West Brom chief executive Ron Gourlay recently revealed that he 'won't be afraid' to sell players this month - and there could be a busy end to the window at The Hawthorns as he is also open to adding to Carlos Corberan's promotion-chasing squad.
The Baggies could be looking to secure deals for South Korea striker Cho Gue-Sung and Brighton striker Deniz Undav.
And Gourlay told the Express & Star: "I never like to sell players, but if we do have to sell players to keep the club running sustainably, then that'll be a decision I'll have to take - I won't be afraid to do it to protect the club."
Grant has just returned to fitness and came off the bench for the final minutes of last Friday's 2-1 defeat at Burnley.
And the £15million man's return is welcome news for Corberan, who is reeling from the news that Phillips' quad injury is worse than feared.
The versatile winger has been a revelation since Corberan replaced Steve Bruce in the Hawthorns hot-seat - but could be facing nearly three months on the sidelines after scans revealed an issue with the tendon in Phillips' muscle.
"The injury to him is a more serious injury than we were expecting," Corberan told the Express & Star. "He is going to go to see a specialist to know more detail the number of weeks that he will be out.
"But we are talking a minimum number of six and maximum number of 12.
"So yes, it's a big injury, a serious injury, we still need the medical confirmation for detail but it's not an injury of one or two weeks."
Phillips has operated predominantly on the left flank so far - and Corberan sees Grant as a left winger, preferring either Daryl Dike or Brandon Thomas-Asante to lead the forward line.
Grady Diangana can also fill in on the left flank but is struggling for consistency - so the Albion coach is more likely to keep Grant in his ranks this month, with the former Huddersfield forward having scored 22 league goals in 68 appearances since moving to Albion in October 2020.