A former teammate and coach for Solskjaer during the Norwegian's time in charge at United, Carrick beat him to the job this time around.
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And the Geordie says Solskjaer has been in touch to congratulate him.
"Yes, I've spoken to him," said Carrick.
"We're close. We've been through a lot together, so he's been fully supportive as you would expect.
"He's some man and I respect him an awful lot. He wished us all the best and was happy that we got the right result."
Carrick knows Solskjaer is "there if ever I need him" but admits he is "not one that bothers a lot of people".