Hearts defender Souttar has signed a four-year contract with Rangers who would have to pay a fee of around £500,000 if they wish to accelerate his arrival during the current transfer window.
Van Bronckhorst has expressed his delight at the capture of the 25-year-old and, as it stands, is operating on the expectation he will join his squad in the summer.
But with Rangers securing the club record £16 million sale of Scotland right-back Nathan Patterson to Everton earlier this month, van Bronckhorst hinted that it provides him with the opportunity to sign Souttar before the end of January.
“There is still a transfer budget we have for this window, so we might add another player to the squad for the balance as well,” said van Bronckhorst.
“If it’s going to happen, I don’t know. We just have to wait and see how it develops. I’m happy John is joining us. He is a very good defender, Scottish international player. I also think it’s very important for us to keep our Scottish identity.
“He fits perfectly in that role. So I’m happy he will join us for the next season. We’ll have to wait to see what will still happen in this window but, as far as I know, we welcome John in the summer.
“We are happy with the move Nathan had. I think it’s a very good move for him because he joins a very strong league at a young age, so I’m really happy for Nathan. Of course, it was also good for the club because of the financial package we will receive.”
Rangers return to action after the rescheduled winter break when they face Aberdeen at Pittodrie on Tuesday night.
In addition to being without Joe Aribo, who is on Africa Cup of Nations duty with Nigeria, van Bronckhorst has further midfield absences in the shape of injured duo Scott Arfield and Steven Davis.
That could mean an immediate debut for USA midfielder James Sands who has joined Rangers on an initial loan deal from New York City.