Konta was 6-3 3-0 ahead when play was suspended until 11am (4pm BST) on Thursday, raising the prospect of a third round clash with world number five Elina Svitolina the same day if Azarenka yields.
Romanian Buzarnescu left her contest against defending champion Elina Svitolina in a wheelchair having suffered a nasty-looking injury when attempting to make a backhand shot from the baseline.
The strongest tennis player of Ukrainian Elina Svitolina won a second victory at the tournament in Montreal category, WTA Premier 5 with prize money of $2,820,000.
Konta, who was leading former world No 1 Azarenka 6-3 3-0 on Wednesday before rain forced an overnight delay, returned to wrap up the match in comprehensive fashion 6-3 6-1.
In the third round of the competition planted under 5-m number Svitolina 1 hour 25 minutes defeated Joann Conti from the United Kingdom - 6:3, 6:4, according to UKRINFORM.
She held her nerve to serve out the match, claiming a 6-3 6-4 win after 85 minutes on court.
In the quarterfinals of the tournament in Montreal Elina Svitolina will play against the victor of the match between the Belarusian Arina Sobolenko and Belgian by Alice Mertens.
The Ukranian needed just one break of serve to seal the first set, and surrendered just three points on her first serve.