Neither of those achievements are guaranteed just yet, with the Red Devils needing to take at least a point from their final two top-flight fixtures in order to wrap up second spot.
Attention will then shift to the pursuit of silverware as Jose Mourinho prepares for another reunion with Chelsea in the FA Cup final.
Giggs believes that progress is being made at Old Trafford - with the club having tasted Community Shield, League Cup and Europa League success last season - and expects them to soon be challenging for the biggest and most prestigious prizes.
A man who collected 13 Premier League titles and two Champions League crowns during his distinguished playing career told PA Sport on United’s class of 2018: “To finish second and win the FA Cup is a successful season.
"If they do that we'll definitely be getting there, back to the top.
"As a United fan you want to compete for the league and compete in Europe as well, but winning the FA Cup and coming second, you can build on that and we all hope they can compete for the Premier League next year.
"They were quite successful last year, winning two trophies, but coming second is definitely progress."
While keeping a close eye on developments at Old Trafford, Giggs is also monitoring star turns elsewhere in his role as manager of the Wales national side.
Among those still competing for major honours this season are Real Madrid forward Gareth Bale and Liverpool youngster Ben Woodburn, with their respective sides set to lock horns in the Champions League final.
"It's huge for Gareth, it could be his fourth winners' medal, which is unbelievable, and for Ben it's a great experience, such a great occasion," Giggs added.
"It can only help Ben in what I believe will be a very successful career. It's about taking it all in, seeing how players prepare for these big games. You're seeing great players at first hand, players at Real Madrid and Liverpool as well, with what they've done recently.
"He can look and pick up pointers from how these players prepare for such a game."