The pair were heavily fined on Thursday for missing Middlesbrough's first training session of the pre-season camp without permission.
The majority of Middlesbrough's key figures have been on the lookout for new deals since their team's demotion to the Championship.
But Southgate insists that Boro's financial demands have to be met in order for any player to depart.
"Both players (Mido and Alves) are our players and we have a financial reality about things. We won't have a fire sale," he was quoted as saying by Gazettelive.
"Any clubs that want to buy any of our players have to match our valuations if they want to do business.
"Mido is our player and, as long as he is, he has to come in for training.
"We have had no bids for any of our players. We expect players under contract here to come in and get fit for the start of the new season."
Sunderland head coach Steve Bruce is reportedly aiming to land Mido in the current transfer window for ₤2.5 million (approximately EGP20 million).
The 26-year-old worked under the management of Bruce in the second half of last season at Wigan Athletic, when the former Manchester United man was in charge at the JJB Stadium.
Mido joined Wigan on a six-month loan deal from Middlesbrough in January, and scored two goals for the Latics.
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