The Egyptian international previously said that he wants to leave Boro, especially after the club's relegation from the English Premiership.
While confirming the bid, Southgate refused to unveil the name of the approaching club, adding that Mido will have the final word on where he will move next.
"We have received an offer for Mido which the club finds acceptable. It's up to the player and maybe something will happen over the next week," he told the club's official website.
According to British media, the 26-year-old could be on his way to Sunderland, where he will rejoin with former Wigan Athletic manager Steve Bruce.
Mido played for Bruce when he was loaned out to the Latics for the final six months of the past season.
However, Zamalek could turn out to be Mido's surprise suitors after the White Knights expressed a desire to bring the player back to Egypt.
The nomad striker showed up late for Middlesbrough's reunion following the summer holidays, a behaviour that prompted the club board to fine him.