The Potters travel to Craven Cottage on Wednesday night where last week Roy Hodgson's side booked their place in the Europa League final with victory over Hamburg.
Both Stoke and Fulham - currently 13th and 11th respectively - are looking to claim a top-half finish in the Premier League this term.
The London club made it into Europe by claiming seventh spot last season and Shawcross feels it would be "fantastic" if the Potters could follow their lead in the next campaign.
"Next season, our first aim will be to reach a points tally where we can't be relegated, and then we will push on from there," Shawcross told the Sentinel.
"I would love to play in Europe - ask any of the Stoke lads and they would say the same.
"It is the ultimate for a professional. If we could do it, it would be fantastic, although it is difficult and the first aim is to get to that safety mark.
"But Fulham are a good example of how, if you get off to a good start, you can push on."
Shawcross was named Stoke's young player of the season on Sunday, while midfielder Matthew Etherington picked up three awards for overall player of the year.
Fabio Capello called Shawcross up to the England squad in February and the defender believes Etherington may have also been given the nod had he not sustained an injury in a match against Manchester City which was being watched by the national coach.
"He has been brilliant this season and it has come as no surprise to me that he has won the awards he has done," Shawcross said.
"Matty is a top lad and a top player, and I'm glad for him that he has won these awards.
"I honestly believe that had he not picked up an injury against Manchester City then he would have been in with a shout of an England call-up, that's how good he has been this season.
"Hopefully, he can continue to progress next year and then who knows, hopefully Stoke could have a couple of players in the England squad."
Etherington - who joined Stoke in January 2009 - was delighted with his individual awards, but paid tribute to the collective endeavours of his team-mates and the club's fans.
"Personally, it is good for me, but there is a squad of players there who have played a massive role in such a successful last few years for this football club," Etherington told stokecityfc.com.
"They deserve huge amounts of credit as well, because as we always say here, it really is a unified effort.
"I include the supporters in that too because they have been remarkable ever since I arrived here.
"When people tell you about them, it is hard to believe some of the things they say, but having been here for 18 months now, they really are special."
Following their promotion from the Championship in 2008, Stoke finished 12th last season and with two games left to play this time out, are currently one point away from matching their points tally of 45. They will move up to 10th if they can better it with victory tomorrow night.
Club captain Abdoulaye Faye remains an injury doubt due to a thigh strain, while Asmir Begovic will continue to deputise in goal for Thomas Sorensen, who is recovering from a dislocated elbow.
Striker Dave Kitson was left out of the squad for Saturday's goalless draw with Everton after he reacted to being substituted in the 7-0 defeat at Chelsea by walking down the tunnel.
Kitson hit back at manager Tony Pulis' comments that he was "petulant" in an interview with the Sunday Times published at the weekend, and his future with the club remains uncertain.