The Democratic Republic of Congo has reportedly convinced Wan-Bissaka to switch international allegiances, with the 23-year-old having played for England's U20 and U21 sides.
While Solskjaer wouldn't tell Southgate to pick Wan-Bissaka, the Norwegian has been completely satisfied with his performances this season, along with fellow England fullback Shaw.
"I don't want to do Gareth [Southgate]'s job," Solskjaer said when asked about Wan-Bissaka's chances of making the Euros squad.
"I feel both my full-backs have done really well. I've been very pleased with them. The development of the two of them, the improvement has been there to see.
"Gareth has got good options, if he fancies. I'm not going to say that he should or shouldn't."