Saints have enjoyed a fine second season back in the Barclays Premier League, with a comfortable top-10 finish all-but secured.
Off the field, though, things have been less serene following the January exit of ambitious - if divisive - executive chairman Nicola Cortese.
The departure raised question marks over the direction of the club and, while the much-speculated meltdown at St Mary's has not come to fruition, talk remains over the futures of star players and manager Mauricio Pochettino.
"I would say it is a big time for the club," highly-rated teenager Shaw said.
"They've got to get the manager down on a long-term contract, which will probably be the first thing they try and do.
"Then, obviously, to keep the stars that have been mentioned in the papers that have been linked with different clubs.
"This is a very important summer and if they want to push on then they are going to do things that I've just said.
"This club can push on, definitely, I think if we get our squad strengthened.
"I think in the next year or so we can be challenging the top five teams."
Owner Katharina Liebherr and her recently-formed board will sit down with Pochettino at the end of the season to discuss the plans going forwards.
Tying the former Argentina international down to a new deal would certainly be a sign of intent, and one which could keep the vultures away from St Mary's.
Adam Lallana and Rickie Lambert have been linked with a move to pastures new, although it is 18-year-old Shaw attracting the most attention.
Manchester United, Chelsea and Manchester City are all reportedly after the England international, who this week was linked with Spanish giants Real Madrid.
Shaw said: "There is not a club I am not linked with, but I don't focus on that.
"My main focus is my football on the pitch and I just want to keep doing that.
"It doesn't really bother me the attention I get. I just stay focused on my game on the pitch. Whatever is said is said.
"Stuff like that is going to happen but my main priority is the games. We've still got six left, so all the focus is on that."