The two-time Champions League-winning manager, who was sacked by Roma in January, is reported to have signed a two-year deal.
The 28-year-old Manchester United defender has been laid low with his latest muscle issue since mid-February.
Mbappe could be unveiled by the Spanish giants in the coming days before Euro 2024.
Harry Maguire and Luke Shaw are hoping to make Southgate’s Euro 2024 squad.