The 42-year-old spoke to his former teammate on Monday morning and has sent in a CV to chairman Ian Lenagan offering to help.
Rush played 107 games at United after joining from Sunderland for £100,000 in 1994.
Since retiring, he has built up a background in coaching, spending nine months at Gateshead last year, where he was briefly caretaker manager.
The striker is keen to help his former club and thinks Lewis, who wants to wait and see how results go before deciding on his future, would be a good choice.
He said: “If Mickey gets the job then brilliant – I think I could strengthen his application.
“I would bring something else to the table and could take on the forwards.
“I hope Mickey gets it, I think it’s the right time for him.
“I have applied as a coach or to assist Mickey in any way. I know there is a vacancy there and I’m thinking, why not me?”
As well as coaching, Rush believes extensive contacts in his native north east would come in very useful.
He pointed to James Marwood, who scored a stunning goal against United for Gateshead in the FA Cup this season, as an example.
He said: “I helped get James Marwood a contract and he’s absolutely flying. These are the type of players Oxford could be looking at.
“There are little gems up here I could get for next to nothing.”
Rush admits getting the top job at United is a long-term aim, but will not be applying to succeed Chris Wilder.
He said: “I think my ambition says I can do it, but my CV is a bit shallow for the No 1 job.
“I would love that in a few years’ time, it’s what I’m working towards. I want to one day take them back to where they should be.
“But I wouldn’t want to go over Mickey’s head and it’s no good running before I can walk.”