Lingard, a United academy graduate, ended his 22-year association with the Old Trafford club following the expiry of his contract last month.
The England international, who made his United debut in 2014, scored 35 goals in more than 200 games for them.
Forest did not specify the length of his contract but Sky Sports reported he had signed a year-long deal.
Lingard had been linked with a switch to Newcastle United as well as a return to West Ham United, where he enjoyed a successful loan spell in the 2020-21 season, scoring nine goals in 16 matches.
He will wear the number 10 jersey at Forest.
Forest have been busy during the transfer window, signing defender Harry Toffolo (26) and midfielder Lewis O'Brien (23) from Championship side Huddersfield Town on Wednesday.
They have also recruited Dean Henderson (25), Taiwo Awoniyi (24), Brandon Aguilera (19), Neco Williams (21), Giulian Biancone (22), Wayne Hennessey (35), Moussa Niakhate (26) and Omar Richards (24).
