McKevitt of Beech Park, Blackrock in County Louth, was convicted in 2003 of directing terrorism.
He was sentenced to 20 years in prison.
Among those reported to be backing the family's lobbying for his release is former Fianna Fail government minister Ã‰amon Ã“ CuÃ­v.
The paper reports the others as independents Clare Daly, Mick Wallace, Maureen O'Sullivan and Thomas Pringle.
McKevitt had a cancerous kidney removed in May and had been temporarily released as a result. He has since been returned to Portlaoise Prison.
McKevitt's son Stephen told the Irish Times the TDs (members of parliament) see it as a "humanitarian issue".
"They agree that my father should be afforded the same rights as anybody else," he said.
In a landmark civil case in 2009, the High Court in Belfast ruled McKevitt and three other men were responsible for the 1998 Omagh bomb that killed 29 people and unborn twins.
The other three men were Liam Campbell, Colm Murphy and Seamus Daly.
McKevitt did not face criminal charges in relation to the Omagh bombing.
Five Irish MPs are supporting the family of Real IRA leader Michael McKevitt as they seek his release from prison while recovering from surgery, the Irish Times is reporting.