The Manchester Evening News says Solskjaer is edging towards Henderson as his number one goalkeeper next season.
Solskjaer has conceded the goalkeeping dynamic between Henderson and David de Gea is unsustainable for a second season and Henderson is currently in line to start the season opener against Leeds United.
Henderson, 24, withdrew from the England squad on Tuesday because of a hip injury and has been given a month off by United, but is expected to be available for a full pre-season. De Gea's return could be delayed due to his participation in the European Championship with Spain.
De Gea, 30, started 36 times for United in 2020-21 and Henderson played in 26 games in his first full season with his parent club.
Henderson had an eight-game run as United's Premier League number one in March and April but De Gea started the Europa League final after lining up in both of the quarter-final and semi-final legs.