The 24-year-old left-back appears to be Steve Cotterill’s choice to fill the void on the left side of defence left by Charlie Daniels, who exited Shrewsbury this week on the expiration of his short-term deal.
Galloway, who can also play as a left-sided centre-half, has made just one appearance for the Championship Hatters this season, coming in the recent FA Cup victory over Reading.
The former Toffees left-back suffered a knee injury at the end of 2019, just a couple of months into his Luton career, that saw him sidelined for over a year.
Having made his MK Dons debut well short of his 16th birthday in the FA Cup in 2011, Galloway was picked up by Everton from MK Dons in 2014, penning a five-year deal, and spent time out on loan in the top flight with West Brom, as well as Sunderland in the second tier.
The defender, who was born in Zimbabwe, was a regular for the Goodison Park outfit in the 2015/16 season, where he made 19 appearances for the senior side. He made 11 appearances for England under-19s and three at England under-21 level.
Galloway was understood to be out of contract at Kenilworth Road at the end of the season.
Town appeared to confirm the deal via social media before the statement was deleted soon afterwards.
It is believed Shrewsbury are working on finalising the permanent deal to bring Galloway to Montgomery Waters Meadow.
He would become the fourth new addition of the January window at Town, following loan duo Harry Chapman and Matthew Pennington and midfielder David Davis, who penned a deal until the end of the season.
Cotterill and his staff are also likely to be looking at a right-sided defender ready to replace Matt Millar, who returned to Australia after a deal could not be reached to make his Shrews stay permanent.