Spurs have a fine record since Sherwood replaced Andre Villas-Boas at the helm in December, winning five games out of six under their new boss to stay in contention for a top-four spot.
However, Spurs remain fifth after Sunday's 3-1 victory away to Swansea and Sherwood is adamant that they still need to improve.
He believes it is essential to book a place in Europe's premier competition for next season, suggesting that he may struggle to keep his job if they miss out.
"If the season was to end tomorrow the chairman wouldn't be too happy because we're fifth," said Sherwood, who was handed an 18-month deal by Daniel Levy after Villas-Boas' departure.
"The final league position has to meet the expectations of the club otherwise it's 'Goodbye Charlie'.
"The club need to finish in fourth place. Anything other than that is going to be a disappointment.
"Realistically we should be in and amongst it. But it ain't easy, is it?"
Sherwood has challenged his Tottenham players to keep their good run going until the end of the season, although he would ideally like some of their rivals to start slipping up as well.
He added: "No one seems to be giving any leeway. I keep looking at results thinking 'Surely they're going to drop points here today'. And they don't.
"They were probably looking at us today thinking 'They might slip up at Swansea.' So we're not giving anything away either.
"It's going to go right down to the wire, it's just about keeping that consistency and belief.
"It probably boils down to how you're going to get on against each other."
Emmanuel Adebayor scored twice in the win at Swansea, and Sherwood was full of praise for the striker after another impressive performance.
"It's great," he told Sky Sports. "It's obviously great for the football club and it's great for Ade himself.
"He's flying at the moment and he's enjoying his football. He's playing with a smile on his face and he's finding the back of the net.
"I thought we played well. Even though Swansea were having some chances from distance in the first period and they knocked it aro