Twenty-four-year-old Cole arrives at the club on a deal until January 2018.
Born in Manchester, he began his career with Manchester United.
He came up through academy at Old Trafford, playing a big part in the club’s run to the FA Youth Cup final.
He went on to sign professional terms with United and made his senior debut in the Carling Cup in September 2011 in a 3-0 victory over Leeds United and won six caps for England at under-19 and under-20 level.
Cole joined Fulham in 2014, for an undisclosed fee, and made his debut in a 2–2 draw against his former club at Old Trafford.
From the Craven Cottage-based outfit, the wide man went out on loan to MK Dons before teaming up with now Tranmere manager Micky Mellon at Shrewsbury Town during the 2015/16 season where he made a total of thirty-four appearances.
Last season, he spent time on loan in the Scottish Premiership with Inverness Caledonian Thistle and left Fulham at the end of last season.
Green (pictured) joins the club for an undisclosed fee from Evo-Stik League Premier side Stourbridge.
The 21-year-old began his career with Burton Albion before Blackburn Rovers snapped him up for an undisclosed fee as a 16-year-old.
He progressed through the academy at Ewood Park before signing a two-year professional contract.
Green spent time with Cardiff City last season before joining Hednesford Town and then the Glassboys this term.
Mellon said: “I’d like to welcome Larnell and Devarn to the club and we are looking forward to working with them both.
“We have been looking for players who can isolate people, give us a bit of width because we want to play with width, and create more chances for the front players.”
Both players are available to make their Tranmere debuts in tomorrow’s home game against FC Halifax Town