Sam Allardyce's men are currently fifth in the table, a point behind Manchester United, who occupy the final UEFA Champions League qualification spot.
West Ham fans were critical of their own side's style of play at times last season, but they have enjoyed a stellar start to 2014-15, aided by some shrewd signings.
The Upton Park club were fifth at the end of the 1998-99 campaign under Harry Redknapp, but have finished no higher than seventh in English football's top tier since then.
Tomkins, who scored the winner in a 2-1 victory at West Brom on Tuesday, believes a top-five finish is an attainable target for the club.
"It's not impossible for West Ham to break the top five," he said.
"Look at our position last season [13th] and now. It just shows a bit of team spirit, good organisation and good players, and you get the benefits.
"The lads are excited about what we're doing and it's all about keeping the run going.
"Our fans are saying we are going to be top of the league but we are enjoying how we're playing. We're also getting a few players back from injury, which makes the team even stronger.
"Who knows what we can achieve? Watch this space because we're on a good run.
"It's hard to get to get to where we are but the hardest part is to stay here.
"Around Christmas you can see a lot of people fading but if we keep our injuries down who knows what people can do?"
