The deals were finalised close to the January transfer deadline, despite being announced after 11:00 GMT, with the players signing contracts until 2017 with the option of a further year.
Both players won the FA Youth Cup with the Reds in 2011 and are promising English midfielders who worked with Cottagers boss Rene Meulensteen at Old Trafford.
Tunnicliffe cut short a successful loan at Ipswich Town in order to seal a return to the Barclays Premier League while Cole has been a regular in Warren Joyce's Under-21s, netting twice in last year's final against Tottenham and finishing the campaign as top scorer for the Reds at that level.
"I've known Rene since I was nine," Tunnicliffe told Fulham's official website. "I'm here now and glad to sign.
"I want to play at the highest level I possibly can and, to me, that's the Premier League because it's the best league in the world. I want to play as many games as I can and improve as a player because I'm only young. We'll see where we go from there."
Cole, 20, completed his routine medical in Manchester and will aim to force himself into the first team at his new club.
Everybody at United wishes the Academy graduates the best of luck with their future careers.