The Yorkshire Evening Post report that Monk has turned to his former club for a transfer which would bring Bartley to Elland Road on a season-long deal.
The defender worked under Monk while United’s boss was manager of the Premier League club and he played in the Championship during previous spells on loan at Birmingham City and Sheffield United.
Birmingham signed Bartley in July 2013 and the 25-year-old played 19 times before Swansea recalled him.
He completed two temporary spells at Bramall Lane in 2010, a time when he was on the books at Arsenal.
Leeds are in the market for new centre-backs after struggling in that area last term and Swansea appear willing to allow Bartley to move north for the whole of the 2016-17 campaign.
Bartley, who joined Swansea from Arsenal in 2012 and is out of contract in a year’s time, has featured in 26 competitive games for Swansea.
He started Monk’s final fixture as manager, a 3-0 defeat to Leicester City last December.
Monk is yet to secure his first signing as Leeds boss but United have lined up the seven-figure purchase of Swedish striker Marcus Antonsson and sources in Wales claim the club are close to landing Bartley.