New Zealander Hape, 26, has secured an early release from his contract with the Bulls and will leave on June 30 after a six-year stint.
The Bulls say they have received an undisclosed fee for granting that request.
Hape looks set to replace England centre Olly Barkley, who made his Bath farewell in Sunday's European Challenge Cup final victory over Worcester. Barkley joins Gloucester later this summer.
The 24-16 win was also saw Bath and England captain Steve Borthwick enjoy a perfect send-off before he heads to Saracens.
And Stevens, an England colleague of Borthwick and Barkley on tour in New Zealand next month, says Bath must now press ahead after securing their first silverware since the 1998 European Cup triumph.
Stevens, a strong candidate to succeed Borthwick as club captain, said: "We don't want to sit on our laurels. We've still got to push the club and push signings.
"People worked hard in recruiting properly for this season, and I think this is now the first step.
"We managed to get some really good signings for this year and next year, so hopefully we can keep on improving, which is the key aspect.
"There was real intent shown by the club this season, and that ambition was rewarded.
"Bath played (on Sunday) like a team that can win finals. We've possibly not had that edge in the past, and hopefully this is the start of great things for the club.
"Winning a final gives us so much confidence going into the summer and then next season.
"What is important is that we don't see this as the end of things for Bath. We must see it as a beginning, a point to build from - a first step, as it were.
"We want to be challenging for the Heineken Cup and everything else that we are in."