Fox Sports Argentina says the 20-year-old is a target for the Red Devils on Deadline Day as Ole Gunnar Solskjaer looks to add a striker to his squad.
United have already failed with a bid for Bournemouth's Josh King but launch a second attempt.
Woodward has now turned his attention to Gaich, who has four goals in nine league matches this season.
The wonderkid has already been capped by Argentina in last year's 4-0 friendly win over Mexico.
However, it's reckoned San Lorenzo have turned down the approach and it remains to be seen if United launch a second bid before tonight.