Bridges told reporters that Ross's health issue was a "very private and personal" matter.
The Botany MP had first raised the "serious" matter with Bridges last week.
"I am, and he is, focused on his wellbeing... on getting better and his young family," the National leader said.
There had been no discussion about Ross resigning as an MP.
"I hope he does come back."
Asked what Ross's return to Parliament would hinge on, Bridges said: "Now it's in his hands to focus on himself and then come back refreshed and well."
National deputy leader Paula Bennett said she was in charge of National's inquiry into the leak and investigator PriceWaterhouseCoopers was still seeking all relevant data from Parliamentary Service.
Bridges said he expected Ross would make himself available to answer any necessary questions as part of the leak inquiry.
"We don't know who is responsible [for the leak]," Bridges told reporters.
Bridges' travel expenses were leaked to media earlier this year. Parliament's Speaker Trevor Mallard announced an independent inquiry, but this was dropped when the leaker revealed they were associated with the National Party and indicated they had mental health issues.
National is continuing with its own inquiry into who leaked details of Bridges spending $113,973 on travel and accommodation between April and June, with MPs asked to sign a privacy waiver.
Ross, 32, said Bridges had been "very understanding and agreed to this along with my request that my transport and infrastructure portfolios be reallocated and that I step back from my front bench role".
"I look forward to returning to Parliament in the future to re-join our caucus."
National MP Nikki Kaye, who took time off from Parliament as she battled breast cancer, said it was a sensitive time and she was thinking of him.
"I have had very serious health issues and I'm thinking for him and feeling for him and his family."
Education Minister, and Leader of the House, Chris Hipkins said it was important for MPs to take time off every once in a while.
Hipkins himself was on Parliamentary leave, following the birth of his second child.
"Sometimes people do a better job as MPs if they take a bit of time out. If he needs to take a bit of time out in order to regroup and do a good job as an MP, he should do so."
Hipkins said he would not get into why Ross might have taken time off and that it was a matter for the National Party
Whether or not he was paid during his time was also a matter for National, but there were no processes in place to stop him being paid.
MPs did not get annual or sick leave, he said.
Ross' transport portfolio will be picked up by Paul Goldsmith and his infrastructure responsibilities by Judith Collins. Paul Goldsmith's revenue portfolio will be picked up by Andrew Bayly, Bridges said.
Ross is the MP for Botany and entered Parliament in 2011.
He is a former Manukau City and Auckland councillor for Howick between 2004 and 2011.
