United's most decorated player was named as interim boss following the departure of David Moyes earlier this month. But while Dutch boss Louis van Gaal is expected to take charge on a permanent basis at the end of the season, the report claims Giggs wants to be considered if results continue to improve.
The 40-year-old has no first-team management experience and only joined Moyes's coaching staff at the beginning of this campaign.
He refused to say he didn't want the job long-term when asked by journalists during his first press conference as temporary Manchester United boss.
And following the 4-0 win over Norwich in Giggs's first game, the clamour for him to be appointed on a permanent basis is growing.
If he gets positive results in the three remaining games before the end of the season, the United board may find it difficult to ignore the fans' calling for Giggs's appointment.
Players like Wayne Rooney and Anders Lindegaard have supported the Welshman since he took charge, but the report notes that behind closed doors United's decision makers remain reluctant to hand over the reigns to someone with such little experience.
That means the manager search is likely to focus on veteran bosses like Carlo Ancelotti and van Gaal, with a view to potentially installing Giggs as permeant boss further down the line.