The MEN understands Henderson, 23, spoke with Solskjaer about the need for more playing time after he made his England debut against the Republic of Ireland last month amid frustration over his omission for both home wins against West Brom and Istanbul Basaksehir.
Solskjaer promised Henderson starts in both ties against Basaksehir but reneged on that pledge after United lost the away leg in shambolic fashion.
Henderson is believed to have stressed his experience to Solskjaer, having started in 36 Premier League games for Sheffield United last term after a full season with The Blades in the Championship.
The academy graduate has reiterated he is desperate to become the No.1 at United but is only prepared to wait so long for that opportunity as regular football remains Henderson's priority.
Henderson signed a new five-year contract with United worth in excess of £100,000-a-week in the summer and stands to become the club's first academy product to establish himself as a seasoned No.1.
United made a long-term investment in Henderson with the sole intention of him becoming their first-choice goalkeeper.
Henderson has started four times for United this season and is understood to be open to a loan move in the January transfer window if he is not given a run in the side ahead of David de Gea.
That would potentially put Solskjaer in the invidious position of having to recall Sergio Romero after he excluded him from the Premier League and Champions League squads.
Romero, 33, was denied a transfer in the summer window and is out of contract at the end of the season, so United must sell him next month if they are to obtain a fee. United could still get a fee if they trigger Romero's one-year extension and decide against releasing him.
Solskjaer's loyalty to De Gea backfired in United's Champions League elimination by RB Leipzig, where he was criticised for his feeble attempt to thwart Justin Kluivert's winner. De Gea, 30, was also culpable for goals against Basaksehir and Southampton.
The MEN revealed this week a number of United players believe Henderson should be starting ahead of the under-pressure De Gea, dropped by Spain coach Luis Enrique for their games against the Netherlands, Switzerland and Germany in November.
Henderson replaced the injured De Gea in United's 3-2 comeback win at Southampton two weeks ago and made his full Premier League debut for the club last week at West Ham, where De Gea was rested ahead of the Leipzig tie.
De Gea's decision to go off at half-time against Southampton was questioned by some at United as he trained the next day at Carrington before lining up for the home defeat to Paris Saint-Germain three days after he was injured.
Henderson is certain to start in United's Carabao Cup quarter-final at Everton on December 23 but there is a growing clamour among supporters for him to line up in Saturday's Manchester derby.