Former Oldham player Richie Wellens has been in interim charge since John Sheridan left last month and the club are 19th in League One, a point above the relegation zone.
Scholes lives in the town and has been spotted at Boundary Park a number of times over the years.
Ex-Netherlands midfielder Clarence Seedorf also held surprise talks with Oldham, but is now out of the running.
He won 11 Premier League titles at Old Trafford and was capped 66 times by England before announcing his retirement in 2004.
"They are a really good family club who mean a great deal to a lot of people - myself included". The club will decide whether to appoint him or Scholes in the next couple of days.
The 42-year-old is also part-owner of Conference North side Salford City along with Giggs and former United teammates Gary Neville, Phil Neville and Nicky Butt.
It would be a hard situation for a managerial debut with Oldham sitting at 19th in League One, and also considering that current caretaker manager, also an ex-United player in Richie Wellens, has done a solid job thus far.
The League Out outfit remain in 19th-place in the league and host Blackburn Rovers on Saturday afternoon.
- Trader's Roundup: Vipshop Holdings Ltd (VIPS)
- Smoking on the beach banned
- Australia jet and navy data stolen in 'extensive' hack
- Smoke from wildfires prompts health caution
- Assassin's Creed Origin DLC Details Revealed
- Federation Internationale de Football Association needs to spend on goal-line technology that could have saved USA
- Lowe's Companies, Inc. (LOW)
- Alina Sheykhet murder: Ex-boyfriend charged in her death
- Razer teases 1 November event, likely for a new smartphone
- Helios & Matheson CEO eyes options for MoviePass expansion