The 20-year-old came through the ranks at Goodison Park but has not yet managed to break into the Toffees' first team.
A fruitful loan spell at Yeovil Town in 2015-16 was followed up by a stint at Birmingham City this season which was curtailed by a change of manager.
Walsh was signed on loan by the Blues in August but failed to feature following the sacking of Harry Redknapp in September and returned to Everton to train last month.
According to The Sun, Walsh's advisers have already met with Robins boss Lee Johnson ahead of a potential move early in the January transfer window.
Walsh is an England youth international and has won caps at Under-16 and U18 level.