Ms Grant said she was "confident" that a backlog of up to 30,000 applications would be cleared in time for families to go away.
But she added that there was a "lot to be said for the 'staycation"'.
Earlier this week Home Secretary Theresa May issued an apology to people whose travel plans had been disrupted by the delays.
HM Passport Office says it has been dealing with the highest demand in 12 years, with the summer surge in applications starting earlier than usual.
Asked about the situation in an interview with the House magazine, Tory MP Ms Grant said: "I'm in no doubt, I'm very confident that people will get their passports. But if they don't want to go away, we have some fantastic places to visit and holiday not that far from here. I think there's a lot to be said for the 'staycation'."
Frustrated Britons caught up in the passport debacle branded Mrs Grant's comments "appalling" and said she was "not living in the real world".
Martin Cook, 43, from Ipswich, endured delays getting his passport and only finally got it the day before he flew out to Prague for a weekend away with his wife Annabel last month, despite submitting his application four weeks earlier.
The BT project manager said of Ms Grant: "Her comments are laughable. It is ludicrous. It is not her place to tell people where to holiday. If somebody has made their plans and put their passport application in in good time they should be able to travel where they like. It is a basic liberty."