The Shropshire Star understands Greene, who has been out of work since leaving Boston in November 2016, has been interviewed for the vacant role.
The Bucks have received around 15 applications for the position to replace Rob Edwards at the New Bucks Head.
Chairman Andy Pryce is now conducting interviews with the shortlist as Telford hope to make a swift appointment as soon as possible to give the chosen boss sufficient time to prepare for pre-season.
Prior to joining The Pilgrims, Greene won back-to-back championships with Cambridgeshire-based St Neots Town, who he took to the Southern Premier Division at Step Three.
Having left St Neots in the summer of 2012, Greene took the job at Conference North Boston the following year. His side finished one place outside the play-offs in their first season, before consecutive semi-final defeats in 2015 and 2016 against Chorley and North Ferriby respectively.
Greene won plaudits for helping the likes of star players Dayle Southwell and Zak Mills into the Football League.
Greene is not the only former Boston chief linked with the role at the New Bucks Head as Adam Murray has also emerged as a candidate.
Ex-Derby, Kidderminster and Oxford midfielder Murray took charge at Boston in October 2016 but resigned a year later after a poor start to the National League North season had the Pilgrims second-bottom.
Murray had previously been in player-charge of League Two Mansfield, where at 33 he was the youngest boss in the Football League. He remained in charge of The Stags in the fourth tier for two years.
Ex-TNS boss Craig Harrison, who won six titles at Park Hall, and former Bucks defender and first-team coach Gavin Cowan and are also vying for the role, with Cowan determined to take his first steps into senior management.
Neil Kitching and ex-caretaker boss John Psaras have also thrown their hat into the ring to replace Edwards, while former Solihull Moors and Grimsby and Chester manager Marcus Bignot – who left Chester last month – is rumoured to be interested.