Spurs impressively overcame Borussia Dortmund 4-0 on aggregate, while United staged a remarkable comeback to beat PSG 3-1 in France on Wednesday and go through on away goals with a 3-3 aggregate score.
Berbatov told Betfair: “No one's going to dare write United off anymore, are they? As the saying goes, 'Respect the giant, otherwise you'll suffer'.
“Their players will have been saying 'we'll take anyone next' in the dressing room on Wednesday, and they're proving week after week that they're a match for anyone.
“Both they and Spurs have a chance of winning the Champions League now - they're in the last eight, so why not?
“Spurs are gaining more experience in the competition every season and they have great attacking talent, with Dele Alli returning soon to complement Kane, Son and Eriksen. A bit of luck in the draw and you never know.”
Turning his attention to the Premier League title race, the Bulgarian underlined the importance of edging narrow wins as Manchester City and Liverpool battle to be named champions, with one point currently separating the two sides.
“It's a nerve-wracking time in the title race for City and Liverpool, and right now Pep's team are handling it a bit better, as you'd expect from a team who has won it before,” he said.
“One goal becomes so important at this time of the season - City have won their last two games 1-0, whereas Liverpool have drawn a few, and are missing chances to win.
“When I won the league with United, we won 10 games 1-0 - it's a real champions' scoreline.
“Mo Salah missed a few chances you'd expect him to score at Everton, and it looked like a sign of nerves kicking in a bit.
“But for all that, nothing is finished. There are still plenty of twists and turns to come and Liverpool are still well in this race."
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