British media have reported the deal could be worth up to 1 billion pounds ($1.39 billion).
Hodgson, who managed England from 2012 to 2016, said he hoped the FA would do what was best for English soccer.
"I think that, if the FA have made a deal, it would be for the right reasons," Hodgson said in comments published by Sky Sports.
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"They see that the amount of money they get from the deal would be advantageous and would be spent in a wise way to help our football... I would hope that games would still be played there by the England national team.
"All I can say is that I am a massive supporter of Wembley as the national stadium, but I have great faith in the FA that they won't be making decisions lightly. If they think this is a good deal, then I will be behind it."
Stoke City manager Paul Lambert also supported the sale, saying it would help improve the atmosphere for England's home matches against smaller teams.
Swansea City boss Carlos Carvalhal, however, was against the deal.
"You can't sell the monuments, in my opinion," Carvalhal said. "If you sell Wembley, do you say you'll sell Big Ben after this? And Buckingham Palace? Wembley is a monument in my opinion and we can't sell the monuments.
"It's culture. We can't sell the culture, in my opinion."
(Reporting by Aditi Prakash in Bengaluru; Editing by Ian Ransom)
