Jim Mallinder’s men square up to the current Aviva Premiership top three before April is out.
Sunday’s trip to table-topping Wasps is followed by a clash with champions Saracens at Stadium MK and a game at Exeter Chiefs.
And with Saints sitting seventh in the standings, every point will be crucial in the pursuit of a top-six spot, which guarantees Champions Cup rugby for next season.
Supporters may fear their side will have a difficult time against the next few opponents.
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But England back row forward Harrison said: “If you want to play at the top for your country, you need to be involved in the big games at a high standard.
“It’s definitely a big chance, especially for some of the boys looking at the Lions tour and others looking at the Argentina tour.
“It’s definitely a big chance for guys to put their hands up to (England boss) Eddie Jones and try to push on from there.
“As a club we want to be contending with the big boys by staying in the top six.
“Ultimately, we’d like to be in the top four. We’ve got the team for it.
“We know we can play well, but we just need to sort a few defensive issues out and we’ll be there.”
So why haven’t Saints quite been there this season?
“It’s one of those things, the bounce of the ball in rugby and you’re on the back foot,” Harrison said.
“We’ve been unlucky with a few mistakes we’ve made and teams just pounce on it to cost us the game.
“Maybe there were a few bonus points we should have picked up along the way but our defence has let us down.
“We’ve got the squad to do well, it’s just those small mistakes letting us down at the minute.”
Saints have struggled to close games out of late, with recent matches against the likes of Bath and Leicester Tigers slipping away towards the end.
And when asked why that is, Harrison said: “In the first place we should be winning comfortably enough to not even have to worry about it, but we need the composure to close out games.
“We can’t afford to get too frantic when it comes down to the last few minutes, rushing passes and making mistakes.
“It’s something we’ll work on, when our finishers come on we want to be adding something rather than just plateauing.
“Our squad’s amazing. Most of the boys have played for their country so we’ve definitely got the team to do it.
“It’s fine margins and we’ll get there.”
Saints will head to Wasps on Sunday hoping to upset the odds.
Dai Young’s men will expect to maintain their impressive home record, which has seen them win their past 16 Premiership matches at the Ricoh Arena.
But Harrison said: “A couple of years back we were underdogs and no one gave us a chance.
“We would show up and get some good wins.
“We’re facing the top of the table and you always want to beat the top of the table.
“They’ve moved up to Coventry so they’re effectively now a derby match and we want to give them a good go.
“We’ll throw our best at them.
“I don’t think there’s emphasis on a big performance. Every game you give your best and every game you give your best performance.
“We want to make top six and to do that we need to start winning games. It starts with Wasps.”
Harrison, who is the reigning players’ and supporters’ player of the year, has been one of the stand-out stars in 2017 for Saints.
“Now it’s about making the next step,” Harrison said. “I need to evolve my game and step it up another level.
“I started the season a bit low on confidence, but it’s slowly come back and being away with England in the autumn and Six Nations has given me a lot of confidence.
“It’s just about trying to put it all out on the field now.”