Manchester United are adamant Ronaldo, who scored 24 goals in all competitions, is not for sale, according to reports, with new manager Erik ten Hag keen to work with him.
Rangnick's arrival brings high hopes as the English club looks to rescue their season.
Solskjaer leaves with United already realistically out of the Premier League title race.
Manchester United's Premier League match against Liverpool is postponed after large number of protesting fans storm pitch as part of protest against club's American owners.
The team will now play England or Australia in the final at Lord's on Sunday.
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