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.
Solskjaer’s first game in charge could be against Cardiff City on Saturday – he was relegated from the Premier League with the Bluebirds during an eight-month spell as manager in 2014.
United will look to appoint a permanent boss at the end of the season.
Former Norway striker Solskjaer has been in charge at Molde since 2015 in what is his second spell as manager. He signed a new contract at the club earlier this month.
As a player, Solskjaer won six Premier League titles and two FA Cups with United.
They are sixth in the Premier League, but are closer to the relegation zone than to leaders Liverpool, who beat them 3-1 on Sunday.