The Daily Mail says Moyes and Martinez top the list, while Frank de Boer of Ajax has also been linked with the job. Former Chelsea manager Andre Villas-Boas is in the running, although as an outsider.
Moyes has established an enviable reputation during his 10 years at Goodison Park. The Scot, who is due to return from holiday on Thursday and plans to take in some games at Euro 2012, would be tempted by the opportunity to manage a club with genuine Champions League credentials.
Martinez's stock has soared after securing Swansea City's promotion to the Championship in 2008 before leaving in 2009 for Wigan, where he orchestrated an incredible escape from relegation last season.
He was a frontrunner to take over at Liverpool last month but eventually lost out to Brendan Rodgers.