There has been no official announcement that the Norwegian is taking over from Jose Mourinho, who was sacked earlier on Tuesday.
But a page on the club's website referred to their ex-striker as "our interim manager". It was later deleted.
Solskjaer, 45, scored 126 goals for United in 11 seasons with the club under Sir Alex Ferguson.
He is in charge at Norwegian side Molde but their 2018 season has finished for the winter and does not re-start until March.
A video on United's official website, featuring Solskjaer scoring the winner in the Champions League final against Bayern Munich in 1999, was captioned: "Solskjaer becomes our interim manager, 20 seasons after clinching the Treble with THAT goal at Camp Nou..."
Prime Minister of Norway Erna Solberg even welcomed Solskjaer's appointment on social media, writing: "Great day for Norwegian football. Good luck keeping control of the Red Devils." She later deleted the tweet.