Allegations about Rooney's private life were made in the News of the World and Sunday Mirror newspapers on Sunday, leading to speculation the Manchester United striker may stay behind.
FA sources have indicated that no decision has yet been taken and as it stands, Rooney is expected on the flight.
Milner explained that Rooney had played a full part in training on Sunday, two days after playing a key role in England beating Bulgaria 4-0, and backed him to play for Fabio Capello's team in Basle.
"He did the full session," Milner said. "We had our full training session and back to football is what we're here to do.
"I think you want the best players out on the field and he's obviously one of them. He showed that the other night.
"He's a top, top player and we want to put out the strongest team possible."
Milner was pressed for more detail about Rooney's frame of mind but stressed: "You're going to have to ask him."
Manchester City midfielder Milner does not expect the claims about Rooney to affect the England squad.
"We've got to make sure they don't," he said. "Things are always said in the paper, good things, bad things, about the England team when results are going well and when they're not going well.
"You have highs, you have lows, and it's down to us to make sure only thing that matters is on the field, come together and make sure we get the result."
Milner indicated England have only the match against the Swiss on their minds.
"Winning football matches is the only thing that matters," he said. "We're here to win a football match.
"We've all trained as a squad in preparation for the game on Tuesday. It's a big game, and it's down to us go out and get the right result."
Asked at a lunchtime press conference how important it is for England players to be liked by supporters, Milner responded: "I think it's important we go out and win football matches and control ourselves on the field and off the field as best we can."
Adam Johnson claims his goal in the rout was the biggest moment of his career to date.
Johnson benefited from some sloppy goalkeeping, but to score was a dream for the 23-year-old Manchester City winger in his competitive debut.
"It was a massive moment," he said. "To score my first goal in my first competitive game, it's the biggest moment of my career, but the most important thing was getting a good win and now we move on to Tuesday (against Switzerland) and hopefully we'll get another three points."
Johnson replaced Theo Walcott, and was able to make the most of a tiring defence.
"When I got on I seemed to have quite a bit of space, especially down the right, and I pushed on to create some more chances and I took my luck and had a go.
"It was just a massive moment. Scoring for your country is obviously something everyone dreams about when you're younger and I'm no different."
Johnson won his third cap for England in the win, but feels he is settling in quickly with the help of several City colleagues.
"I'm getting to know the lads better and better even with some new faces around, but there are a lot of City boys, so it's easier to settle in," said Johnson, who singled out goalkeeper Joe Hart as a player he has known since they played together for the Under-19s.
"I'm good mates with Joe, we're of a similar age and both at City," he said. "We grew up in the Under-19s where we always tried to help each other with stuff. He's relatively experienced now and he's been good just like everyone has."
Hart got to go to the World Cup, where Johnson, part of Capello's preliminary 30-man squad, was left behind.
Although England's campaign ended in a huge disappointment, Johnson said there was no way he could be happy about having missed it.
"A World Cup comes around every four years, and no matter how it went it would have been something to say you went to a World Cup, so I would still like to have gone," he said.
"Now I've got to wait another four years to have another chance. It was one of those decisions, it was disappointing, but it's behind me now."