The Scottish champions jetted off with two of their key title-winning squad, Cameron Carter-Vickers and Jota, signed on permanent deals following impressive loan campaigns.
Argentinian left-back Alexandro Bernabei and former Dundee United goalkeeper Benjamin Siegrist have also bolstered Ange Postecoglou’s squad ahead of a season in which Celtic return to the Champions League group stages following a four-year absence.
McGregor said: “I know the manager was really keen on getting the boys here from last year and having spoken to them over the summer, they were desperate to come back. It’s a huge club and the boys are buzzing to be part of it.
“The club have done their business early and hopefully we can add one or two more faces as well, which would be great.”
Celtic take on Wiener Viktoria on Wednesday before facing Rapid Vienna three days later and Banik Ostrava the following Wednesday.
McGregor said: “It’s an exciting feeling coming back for the new season. We have got some big signings in early doors as well, which is great, good for the group, some new faces as well.
“We are really looking forward to the pre-season trip to Austria. It gives us the chance to go to a remote location and get some really good work done. It lets the lads live with each other for 10-11 days and start to bond and be together.
“Pre-season is a really vital time of the season. You are getting your football fitness in, the manager has got a chance to work with you tactically, individually and with the group as well, what he wants from the team and the intensity he wants us to play with.
“This season we have to go again and reach another level. That’s the challenge for the players and I’m sure the manager will be setting that challenge as well.”
French defender Christopher Jullien, who appears free to leave under Postecoglou, was also part of the Celtic travelling party despite a move to Schalke 04 falling through last week.