The Red Devils have made just one signing so far this summer, with Netherlands international Donny van de Beek completing a £40million move from Ajax.
Ole Gunnar Solskjaer still has over three weeks to bolster his squad further, however, and has been told to sign Bale and Grealish by Sherwood.
Wales superstar Bale is looking for a move away from Real Madrid, having fallen out of favour under Zinedine Zidane, while Grealish has been linked with a move to Old Trafford after impressing last season.
Speaking on Sky Sports, Sherwood said: ‘When teams back off, that’s when Manchester United have struggled, but that was until [Bruno] Fernandes came in.
‘Fernandes could find the key to the door and open teams up. Van de Beek can do that as well, he makes runs and scores goals.
‘But I would also go for Jack Grealish, he can open the door, he’s another Fernandes and gives them another quality that they need.
‘I would also go out and sign Gareth Bale on a free transfer. For me, he needs to come back to the Premier League and start playing football again.
‘I think he has a lot to offer and I think he will flourish for a couple of seasons if he goes to Manchester United.’
Solskjaer, meanwhile, insists Manchester United will always look to improve their squad and will ‘try to make some business’ before the summer window closes if the right deals present themselves.
It’s been a strange time and a strange world. Financially, I’m sure loads of clubs have suffered,’ the Red Devils boss said.
It looks like it as there has not been a lot of activity in the window. We’re always looking to improve the squad, scouting players and, if something fits, both the financial bit and the football bit, of course we’ll try to make some business.
But we can’t know how the markets and how other clubs have been affected either.‘