Vaughan is due to have a fourth operation on his right knee Monday, ruling him out of the remainder of the English season and placing a huge question mark against his participation in the Ashes series in Australia later this year.
Strauss is currently leading the team in their ongoing One-day series with Sri Lanka.
He remains one of the favourites to get the job with Andrew Flintoff, who captained the side in Vaughan's absence in India and during the 1-1 drawn Test campaign against Sri Lanka, out with an ankle problem that means the all-rounder may not play in the first Test against Pakistan starting at Lord's on July 13.
Marcus Trescothick has also deputised for Vaughan but, earlier this week, Strauss's fellow opener said of his captaincy prospects: "Maybe the team has moved on and the next generation is being formed."
There was no such diffidence from Strauss.
"I've said before it's a massive honour to captain England and if the job was offered to me I'd do it," he told reporters at Headingley here Friday.
"I haven't had any consultation with the England selectors at this stage. I'm sure it's something that will need to be resolved sooner rather than later and I'm sure by the beginning of next week something will be ready to be said."
The former Middlesex captain has had a tough time leading England, losing five matches in a row, including a Twenty20, with the team on the brink of a 5-0 One-day series whitewash against Sri Lanka heading into Saturday's Headingley climax.
"It's been tough but I still maintain you learn a lot more about yourself and others when things aren't going well," Strauss said. "It's not ideal to have stand-in captains and I have been a stand-in for a stand-in (Flintoff) which is far from ideal."
Strauss refused to look ahead to England's Ashes defence, saying it was possible Vaughan could yet return to the team.
"Anyone who rules Michael Vaughan out of making a return to the England side would be wrong. Knowing the character he is and knowing how much he still wants to play for England, I think it would be very wrong for anyone to view the side as moving on from Michael Vaughan."
Asked why he wanted a job that has proved a poison chalice in the past, Strauss said: "Captaining any side is a tough job, which is why not everyone wants to do it and captaining England is an even tougher job.
"But it's a great challenge and, if things do go well, there is a massive upside to it as well."
And of his approach to captaincy, Strauss, 29, said: "My personal leadership style has always been to lead by example. You can't ask anyone to do anything you are not willing to do.
"If we want to prepare ourselves properly, that's got to come right from the top. I'm a big believer in leaving as little to chance as possible.
"It's never going to be a slight on me if someone else was selected to do the job but, at the same time, I do feel I've got something to offer.
England are currently without several injured players, both Ashley Giles (hip) and Simon Jones (knee) recently undergoing major operations.
"I suppose they've lost a little bit of direction, not just in their lines and lengths," Sri Lanka coach Tom Moody said of England.
"But I don't think that you can ignore the fact that we have put England under pressure," the Australian added.