The 63-year-old man thought to be Neville Neville was bailed until May pending further police enquiries.
Mr Neville's two sons Gary and Phil both played football for Manchester United and England while daughter Tracey played netball for England.
He is a former director of Bury Football Club, where wife Jill is still club secretary.
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The 10K and half-marathon races take place almost exactly a year after Salman Abedi killed 22 in his attack at an Ariana Grande concert.
Dmitrios Pagourtzis also had several kinds of homemade explosive devices but they were not capable of detonating, county officials said.
After early mist clears, most parts of England and Wales will have another fine, warm day with plenty of sunshine.