The Red Devils are in eighth place in the Premier League table after successive top-flight victories to end their losing run.
Manchester United lost 3-2 to Brighton and Hove Albion before a 3-0 loss to Tottenham Hotspur at Old Trafford.
The 20-time English champions bounced back with a 2-0 win at Burnley before a 2-1 victory over Watford at Vicarage Road.
The Red Devils finished in second place in the Premier League table to secure their best finish since 2013.
Manchester United have been discounted as potential title challengers this season despite pushing Manchester City closest last term.
Former Leicester midfielder Savage believes Arsenal and Spurs will challenge Manchester United for a fourth-placed finish.
“It’s up to Arsenal and Spurs to try and get in the top four ahead of Man United,” Savage told BBC Radio 5 Live.
Manchester United signed Fred, Diogo Dalot and Lee Grant in the summer transfer window to improve Jose Mourinho’s squad depth.
The Red Devils will be hoping to end their title drought following their most recent top-flight success under Sir Alex Ferguson in 2013.
Arsenal have finished behind Tottenham in the two Premier League seasons.
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