Leader of Rugby Borough Council, Craig Humphrey, said Sir Peter was offered the position for 12 months.
Mr Humphrey said Sir Peter, who owns Coventry Airport, would "take the LEP forward".
But concern has been raised about a possible conflict of interest when it comes to development of the airport.
The role has become available following Denys Shortt's recent resignation.
The LEP hopes to create 14,000 jobs by developing the land around the aerodrome and nearby Whitley as part of their Gateway project.
The former leader of Coventry City Council, Conservative councillor Ken Taylor, said the "well-connected" businessman would have to declare an interest.
Mr Taylor said: "We've failed already on this particular project once when we put it to government.
"I don't want that to happen again on the basis that they feel the chairman has got some interest in it."
Conservative Councillor Humphrey said: "There is of course a conflict of interest but I don't think that should preclude Sir Peter from taking the reins of the LEP.
"Taking to one side the Gateway and the airport site, Sir Peter Rigby heads up an organisation that collectively is the fifth largest privately-owned business in the UK.
"He's got a wealth of experience and I think he's the right person to take the LEP forward."
Mr Humphrey added that Sir Peter had asked for two weeks to consider his decision.
Sir Peter Rigby has been asked to be the chair of Coventry and Warwickshire Local Enterprise Partnership (LEP), a fellow board member has said.