The club chairman is keen for a return to Europe and would be delighted if that came via winning silverware.
He confirmed his ongoing ambitions for the club after announcing a £54.4 million loss for the 2024-25 season.
The figures include most of a big-spending summer of 2024 and follow two years in which the club recorded profits totalling about £200 million.
Albion go to Manchester United in the third round of the FA Cup on Sunday.
Four points from two games over the past week, including a draw at Manchester City, have kept them in the midst of a highly competitive race for European places via Premier League finishing position.
Bloom said: “We enjoyed our time in European competition and aim to push regularly for a European spot — and, of course, we dream of winning the first major trophy in our history.
“Easier said than done, but we will keep striving.
“Whatever unfolds by the end of May, I’m confident we’re in for more memorable moments as we continue the next stage of our journey together.”
Albion head coach Fabian Hurzeler shares his chairman’s targets.
Speaking ahead of the Old Trafford trip, Hurzeler said: “We always say that we are a club with big ambitions, especially our owner.
“He really tries to live it and therefore we have to fulfil these kinds of expectations, but that is something we love to do.
“We go there, we try to give our best and then we see what happens.”