Spurs have been in the hunt for third place all season but lost twice in March and drew against Everton on Sunday to drop to fourth, with fifth-placed Arsenal now breathing down their neck just two points behind.
"It's been tense. I've felt like that's kind of what the atmosphere's been. That's the way it's been the last few games," Dempsey was quoted as saying in British media on Tuesday.
"It's been noticeable in the last three home games," the American added.
"Who's to say the reason why? Maybe a few injuries, maybe a lot of games. It makes it more interesting for people to watch. It's a little bit more stressful but it's a good stress."
There was welcome injury news for Spurs boss Andre Villas-Boas on Monday, however, when central defender Younes Kaboul, out with a knee injury since the first game of the season in August, completed an hour for the under-21s.
Dempsey said Tottenham's position was preferable to battling relegation or having nothing to play for by being stuck in mid-table.
"At least we're pushing to try to get that (Champions League) spot. Hopefully we have enough character to get the job done.
"I thought the boys showed a lot of character to fight back and get a point against Everton. It's an important point. You would have like to have got three but at least we fought back.
"We have the quality. It's about being confident. It will be difficult all the way until the end of the season."
Tottenham's next game is a Europa League quarter-final second leg tie away to Swiss club Basel on Thursday, the first leg was a 2-2 draw, before they take on second-placed Manchester City in the league on April 21.