Tuttosport says both tabled €65m (£58m) offers for Skriniar.
The Croatian international has previously been linked with both clubs but stayed put at the San Siro.
They are now said to have revisited their interest in his services, with both clubs making an official offer.
It is claimed that United were first to offer Inter a fee, with City matching their bid.
The 25-year-old has struggled to adapt to Antonio Conte's favoured back three formation and, as a result, the club are open to selling him, provided there is an offer which matches their demands.