The Swiss maestro, who last held the top ranking in November 2012, will return to the summit if he reaches the semi-finals at the ATP 500 event.
After seeing off Ruben Bemelmans in the first round, two-time champion Federer came through 7-6 (10-8) 7-5 against Philipp Kohlschreiber in a hard-fought battle on Thursday.
That secured his place in the last eight, before Haase overcame fellow Dutchman Tallon Griekspoor 6-4 6-0, fighting back from a break down in the opener and rescuing three break points in the first game of the second before cruising through.
Federer in full flight
Getting closer and closer. #abnamrowtt pic.twitter.com/GLzNlFWJad
— ABN AMRO WTT (@abnamrowtt) February 15, 2018
Second seed Grigor Dimitrov also enjoyed safe passage although, like Federer, he had to fight to oust Filip Krajinovic 7-6 (7-4) 7-5.
Dimitrov will meet Andrey Rublev in the quarters after the Russian got the better of Damir Dzumhur 6-4 7-6 (7-4), while Daniil Medvedev came from a set down to defeat Pierre-Hugues Herbert 3-6 7-6 (7-2) 6-4 to set up a clash with Andreas Seppi.
The quarter-finals are scheduled to take place on Friday, when all eyes will be trained on Federer as the 20-time grand slam champion bids to create yet another slice of history.
Photo Getty Images. Caption: Roger Federer