The publication hopes the Stones might be persuaded to repeat their appearance at the 1964 NME Poll-Winners' All-Star concert where they joined the "greatest array of pop talent in the world", including The Beatles, at the event, which was hosted by Jimmy Savile.
The Stones, who could "hardly be heard above the shouts" at the Empire Pool, Wembley, played three songs and took the award for Best Rhythm and Blues act. Mick Jagger was named Best New TV Singer.
Mick Jagger said: "It's very nice to be nominated. When I first heard it was four things, I thought, 'Ooh, blimey! That's very nice!' And it's not a journalists-only thing, it's not a coterie of closely-knit people choosing."
Mick (69), added: "It's funny, because when we were rehearsing at Wembley arena last year, it was where we used to do the NME Poll-Winners. We remembered, it was the first time we ever played Jumping Jack Flash, at one of those Poll-Winners concerts." David Bowie (66), is also back in favour after his surprise return with a new song after a decade of semi-retirement. Bowie, who won two NME awards in1974, is nominated for Music Moment, Best Video and Hero of the Year.
Pussy Riot, the Russian punk group whose members were imprisoned following their anti-Putin protest, are nominated for the "punk prayer" they performed at Moscow's Christ the Saviour cathedral last year, which prompted their arrest.