The Gunners had a £30million bid to sign the Netherlands international rejected earlier in the week.
Their offer is believed to fall £20m short of Ajax's valuation of the 22-year-old, who is capable of playing as a centre-back or right-back.
Though talkSPORT understands that the Gunners are hopeful an agreement will be reached.
Timber is also believed to be keen to make the move to the Emirates - a year after he almost joined Manchester United.
And now Jurrien's sibling Quinten, who recently helped Feyenoord pip Ajax to the Eredivisie title, has reportedly confirmed his brother is on Arsenal's radar.
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After captaining the Netherlands' Under-21 side to a 0-0 draw against Belgium in their European Championship opener, Quinten was asked about Jurrien's future.
Speaking to Dutch news outlet NOS, he was asked whether Arsenal's interest in his brother is correct, where he is alleged to have said yes.
He then added of Arsenal's opening proposal: "I don't know if there was an offer."
And Quentin then conceded that he is unsure whether Jurrien will be playing in the Eredivisie next campaign, as he stated: "I don't know."
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The midfielder also denied rumours that his brother is currently in England after fans claimed he had touched down in the capital ahead of a potential move.
Addressing the talk, he remarked: "He's not in London right now."
It is expected that Arsenal will return with a second offer for Timber in the near future as Mikel Arteta looks to get his business done early before next month's tour in the United States.
A move to sign Chelsea's Kai Havertz in a deal rising to £65million could be completed by the end of the week.
Meanwhile a third bid is likely to go in for Declan Rice as West Ham hold out for £100m.