The same media outlet has claimed that Celtic have a concrete interest in him, but the Premier League duo have been watching him for much of the season, well aware that the 24-year-old has been one of the outstanding players in League One this season.
Villa and Stoke admire him, but the report does claim that they have yet to make a move for Sawyers, who is rated at £2 million - although he could be signed on a pre-contract agreement in January, with his contract expiring at the end of this season.
Celtic are said to be readying a move for Sawyers, and if Villa and Stoke really do want him, they are likely to step up their interest in January.
Who is Romaine Sawyers?
Sawyers, 24, is a former product of West Brom’s youth academy, who has been with Walsall since the summer of 2013.
This season, the midfielder has been in fantastic form, scoring six goals and registering eight assists in 27 games across all competitions.
He has been one of the best players in League One this term, and it is no surprise that his form has seen him scouted by a number of the top teams in England and Scotland.