Bakayoko, 22, has supposedly caught the eye of a number of scouts recently after impressing in Ligue 1.
The France U21 ace was linked to United last month, with reports that he is valued at £10.3million.
Italian newspaper Tuttosport claimed Juventus had watched Bakayoko three times over the summer and that United were aware of rival interest in the burgeoning star.
Bakayoko’s agent even confirmed the two clubs were keeping tabs, along with West Brom and Stoke.
But Italian outlet Calciomercato claim both Juve and United could miss out on the former Rennes player as Milan and Saints enter the mix.
According to the report, Milan will closely examine Bakayoko in the coming weeks before deciding whether or not to make a bid.
Monaco are away from home in each of their next three games, with trips to Metz and Toulouse lined up before a Champions League clash at CSKA Moscow.
Southampton, meanwhile, did not replace midfielder Victor Wanyama this summer when the Kenyan mode his move to Tottenham.
The south coast club could therefore be tempted to make an offer in order to add strength to manager Claude Puel’s midfield.
As for United, they have also been linked with Monaco full-back Fabinho in recent weeks.