South Wales Police said David Ellis, 40, his lodger, has been arrested in Ireland.
Officers are liaising with Garda officers in connection with his extradition back to the UK.
Retired telecoms engineer Mr Warburton, 59, has not been seen since 31 July.
Police launched an appeal for information after he went missing; they said his M-reg Peugeot 205 had been spotted in north Wales before returning to Swansea.
The car was found abandoned at Birkenhead port some days later.
Police said Mr Ellis, who had been Mr Warburton's lodger in Sketty, had stepped off a ferry in Belfast, Northern Ireland, on 6 August.
That discovery prompted a fresh appeal for information and led to Welsh officers being sent across the Irish Sea.
South Wales Police said Mr Warburton's family had been informed of the latest development in the investigation and continued to be assisted by family liaison officers.
Mr Warburton's body has not been found; detectives have previously said they are treating the incident as murder.
Police investigating the murder of missing Swansea man Alec Warburton say a man they wanted to speak to in connection with his disappearance has been arrested.