The fullback joined Saints on-loan in January before signing outright last month.
Walker-Peters believes the capabilities of manager Ralph Hasenhuttl's squad can push them higher up the table next season, after an 11th-placed finish in 2019-20.
He told talkSPORT: “I think the way we finished the previous season, I don't see why we shouldn't be looking to get into the top 10. We finished 11th, we just got above Everton.
“We need to be striving to get into that top 10, continue how we finished the last season and who knows?
“We weren't too far off Europe. We don't want to get too far ahead of ourselves but I think it's a realistic target."
On joining up with the St Mary's side permanently, Londoner Walker-Peters said: “I am over the moon.
“At Tottenham I always found it hard to even get a run of games and now I felt like after lockdown I was given that opportunity from Ralph. I was able to show the qualities I have.
“With a run of games you get confidence and you get your fitness, which is important. Now I am just looking to kick on again.
“My spot in the team isn't solidified, I just need to keep showing what I'm capable of, keep performing and hopefully it will be a great season for us.
“I think we showed after lockdown we have taken a big step in improving as a team. That's what really made it clear that it was the team for me.
“I was in the team for all the games and I felt confident and the team showed we had come a long way."