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Murray, who anchors 5 Live's weekend sport coverage and presents radio quiz Fighting Talk, has impressed BBC Sport executives, according to The Guardian.
The former Radio 1 DJ, who has also hosted football coverage on Five, is believed to have the right sense of humour for the Sunday night football show. He may be drafted in early for the BBC's World Cup coverage, presenting a nightly highlights programme.
Mark 'Chappers' Chapman is another presenter linked to the vacant role.
Gaby Logan will present this week's Match Of The Day 2, while Gary Lineker was already lined up for the final two weeks of the season because of the number of significant fixtures played on a Sunday.
> Adrian Chiles quits BBC For ITV
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