The White Hart Lane club sit level on points with fourth-placed Liverpool in the Premier League after recording their fifth straight top-flight away victory at Swansea City on Sunday.
|9/4||Tottenham are 9/4 with William Hill to finish in the top four|
Asked whether fourth place remained the brief for the campaign, as it was with his predecessor, Sherwood replied: "Yes, the club need to finish in fourth place. Anything other than that is going to be a disappointment.
"If the season were to end tomorrow, I don't think the chairman would be too happy because we're fifth. The final league position has to meet the expectation of the club, otherwise it's 'goodbye Charlie'."
Tottenham have not featured in Europe's premier club competition since the 2010-11 season, when they reached the quarter-finals under Harry Redknapp.
Sherwood believes his side should be one of the sides competing for a Champions League berth, but conceded the form of sides around them makes doing so a difficult proposition.
"Realistically, we should be in and amongst it but it's not easy is it? There's a lot of top teams in there and no-one seems to be giving any leeway.
"I keep looking at the results thinking 'surely they're going to drop points here today' and they don't seem to be doing it. It's going to go right down to the wire - it's about keeping that consistency and belief."