United are expected to release an official statement in the coming days.
Giggs, 42, had a year remaining on his contract as assistant manager but new boss Jose Mourinho intends to fill that role with long-time friend Rui Faria.
Unable to reach agreement over another role at the Premier League club, Giggs, who made a record 963 appearances for United, has decided to move on.
Cardiff-born Giggs has his Uefa Pro Licence, a mandatory qualification for managing in the Premier League, and has never made any secret of his desire to move into management.
Linked with Swansea last season, he felt he was ready to manage United, having served his apprenticeship under David Moyes and then Louis van Gaal.