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Carver was assistant manager during Robson's five-year spell in charge of Newcastle, and was previously the club's centre of excellence director.
He is currently coaching at Leeds and is expected to continue in that role even if he accepts the Republic post.
Staunton, Walsall's assistant boss, is poised to be given a four-year contract to succeed Brian Kerr, who left the post after the team failed to qualify for the World Cup finals.
However, the Football Association of Ireland claim the appointment process for a new manager is ongoing and a announcement is not expected to be made for 10 days.
Former Celtic boss Martin O'Neill was reportedly the FAI's first choice to take over from Kerr - but he is taking a sabbatical from football while his wife recovers from illness.
This has opened the door for Staunton - the Republic's most-capped player, with 106 appearances - to take over.
Former Republic boss Jack Charlton is delighted at the prospect of Staunton and Robson teaming up.
He said: "I think the partnership would be excellent. They would be good together. Robson knows exactly what to do and Steve knows what is necessary.
"Robson has got the experience to teach Steve and his enthusiasm is second to none.
"Steve Staunton is a good lad. I don't know what qualifications he has got for the job, apart from being a good player but I think he would be OK."
It is thought the FAI wants to compensate for Staunton's lack of managerial experience by appointing a mentor - and Robson fits the bill perfectly .