British tabloid The Sun published pictures of Manchester United skipper Rooney, 31, supposedly looking the worse for wear after "gatecrashing" a wedding last weekend.
Southgate believes the claims merit investigation, but he is still waiting to hear from the Football Association about whether he has landed the role full-time following a four-game caretaker stint.
"Over the last few years, the players have been allowed to go home during a period of time," Southgate told reporters after England's 2-2 friendly draw with Spain at Wembley.
"They trained Saturday morning and they trained again on Sunday afternoon, and I gave the players a period of time off (in between). I'm not aware of anything else.
"There were lots of changes we made to routines. Some things we thought we should keep the same and I'll have to review (it) -- or maybe I won't have to review! Someone will have to review it."
Rooney missed the game against Spain on Tuesday due to a knee problem sustained during last Friday's 3-0 win over Scotland in World Cup qualifying.
He is said to have joined revellers celebrating a wedding at England's team hotel in Watford, north of London, on Saturday night and The Sun quoted witnesses who said he was drunk.
Rooney's spokesman said: "As he has always been, Wayne was happy to sign (autographs), pose for photos and chat with guests.
"It is sad that one or two of them have now sought to turn Wayne's friendly good nature to their advantage."
Rooney was recalled by Southgate for the win over Scotland, having been dropped for England's goalless draw in Slovenia, after winning back his place in the United first team.
But his status with both club and country continues to appear uncertain as concerns grow over his waning powers.
Southgate also faces uncertainty over his future, in the short term, as he waits to learn if the FA will offer him the England manager's job.
Whilst he has been at pains not to tout himself for the position, he admits the experience of leading England for four games has given him a taste for managing at elite level.
"I'm proud of what I've done and nobody will ever take that away from me," said Southgate, who was promoted from England Under-21s coach following Sam Allardyce's departure in September.
"I am determined to manage at the highest level possible -- club, country, home or abroad. Whatever it may be. Experiences like this are a big part of that journey."
Southgate last managed in the Premier League in 2009, when he left Middlesbrough after three years in charge.
Asked if he harboured ambitions of working in the Premier League again, he said: "That is also not my decision. I want to manage big matches and I have proved I can do that.
"Wherever it is going to be, I would like to be coaching for a period of time. That is going to require some different challenges at different times I guess."