The Celtic star saw his side crash out of the Scottish Cup to Aberdeen last weekend as Neil Lennon's side were left with only the SPFL Premiership title to play for this term.
The Hoops were dumped from the League Cup by Morton and then finished bottom of their Champions League group after coming up short against Barcelona, AC Milan and Ajax.
Celtic will complete three-in-a-row in a matter of weeks and Mulgrew insists a seat at Europe's top table means more than domestic glory.
He said: "The qualifiers are not easy. I don't think it's just about settling to get into the Champions League.
"You want to go further but you realise that it is hard and with the money that teams are spending in the Champions League, you've also got to be realistic.
"If you were offered the Champions League or a cup, I'd take the Champions League all day.
"I'd rather be in the Champions League and be competing in it every year. If that meant taking away a cup then I'd take that.
"Of course you want to aim high and want to do better - and we should have done better in the cups - but as I've said getting into the Champions League is a huge thing for us."
Celtic overcame Cliftonville, Elfsborg and Shakter Karagandy to book their place in the Champions League this term and will again face a series of qualifiers before they can join the continent's elite next season.
Lennon's side are likely to kick off their European campaign at Murrayfield due to Parkhead being utilised for the Commonwealth Games and Mulgrew hopes the change of venue doesn't impact on their Euro ambitions.
He said: "It's not ideal but if they need to do it they need to do it.
"If it comes to a situation like the Karagandy game, hopefully the fans will get behind us at Murrayfield and travel there to make it a big atmosphere.
"The negatives would be just be not being at Parkhead when you're used to it.
"We'll need to see if and when it happens but obviously we'd prefer it to be at Parkhead.
"I don't know if the crowd will be a big factor.
"The Karagandy game was massive - especially after being 2-0 down from the first-leg.
"The manager gave us a good game-plan and we believed that if we stuck to that it would turn out okay and it did.
"If you did go out, it's not the end of the world as there's still lots to play for but it is massive for the club and the players. Everyone wants to be involved at that level."