Rangers were held 1-1 at St. Mirren to drop points for the third time in five league games, while Celtic came from behind to beat Ross County 2-1.
"We're going to sit down and speak about the last couple of months and which way to go forward," said Van Bronckhorst when asked if he will have talks with the Rangers board in the coming weeks.
"I have no fear, I never have fear, I just see what comes."
Patience with the former Dutch captain has worn thin among the fanbase in recent weeks despite leading Rangers to just a second European final in 50 years last season.
Pressure intensified on Van Bronckhorst after Celtic moved nine points clear at the top despite a shaky start against Ross County.
The Hoops were the victims of a controversial penalty award when Ross County took a shock lead at Parkhead five minutes into the second half.
Matt O'Riley was punished by a VAR review for handball and David Cancola rolled in the resulting penalty.
But two goals in six minutes turned the game around as David Turnbull and Sead Haksabanovic produced tidy finishes from two Reo Hatate passes to clinch an eighth consecutive league win for the defending champions.
"It was a big win and a strong performance. I can't speak highly enough about this group of players," said Celtic boss Ange Postecoglou.
"They were really strong mentally today. The crowd sense it, the opposition sense it - if something goes against us, we're not going to be bowed.
"If anything the boys just put the bit between their teeth and go. It's a great trait to have. Whatever challenge has been thrown at us in the league, we've been outstanding."
Elsewhere, Livingston moved up to third despite conceding a 97th minute equaliser to 10-man Hearts in a 1-1 draw at Tynecastle.
Hibs manager Lee Johnson is also under pressure after a 1-0 defeat at Kilmarnock, while St Johnstone and Motherwell played out a 1-1 draw.
