The France captain is expected to join Los Blancos at the end of his Paris Saint-Germain contract this summer.
Mbappe could form an all-star attack alongside Vinicius Junior, with Madrid's top goalscorer this season, Bellingham, offering support.
"I don't pick the team and I certainly don't make the transfers, but Kylian Mbappe, what a player, who wouldn't want to play with someone as good as him?", Bellingham told reporters at a media day before the Champions League final on Saturday.
"I don't want to put any more pressure on the situation, I know for him it's probably difficult having everyone always talking about him, but it would be really nice."
Real Madrid face Borussia Dortmund at Wembley aiming to win the competition for a record-extending 15th time.
Lifting the Champions League trophy is one of the challenges Mbappe, 25, has on his to-do list.
Mbappe led PSG to the French Cup last weekend in his final match for the Ligue 1 champions and said his next destination will be revealed soon.
"All I wanted was to end things well with my club, with a trophy. I think there is a time for everything and I will announce my future club in good time," said Mbappe.
"I think it will be in a few days so there is no problem.
"I don't know when yet. There are still some details (to be sorted out) but the most important thing was to finish on a high here."