The Red Devils have had an average defensive record this campaign and the goalkeeper has also been inconsistent.
Onana has been guilty of making several errors leading to goals and he could be offloaded during the next transfer window.
As per Fabrizio Romano, United have made contact with the representatives of Vanja, who was previously at the club between 2014 and 2015.
The 28-year-old is open to rejoining the Red Devils but he is likely to demand guarantee playing time next season.
Vanja has been brilliant for Torino in Serie A this campaign. He has kept an outstanding 16 clean sheets from 35 appearances.
Torino are likely to demand around €20 million for the shot-stopper, who is also on the radar of several Serie A giants.
The Turin outfit could prefer to do business with the Red Devils rather than bolstering a direct league rival next season.