The Gunners have already seen a £3m bid for the defender rejected, according to reports.
The Sun reports that the Gunners had previously seen a £3m offer turned down for the promising defender, but with Premier League rivals Liverpool and Tottenham both considering a move for the 19-year-old, it appears that Arsene Wenger is determined to complete a deal before the window closes.
Chilwell has yet to feature for the Foxes in the top flight this season - which is perhaps understandable given Claudio Ranieri's side's current position at the top of the table - but has featured in both the Capital One Cup and FA Cup.
He also enjoyed a successful loan spell in the Championship with Huddersfield Town, making eight league appearances for the Terriers before returning to Leicester at the start of this month.
With Nacho Monreal doing so well currently for Arsene Wenger's side, Chilwell is unlikely to earn a starting spot immediately should he move to the Emirates. However, with the Spaniard now 29, Wenger may well view the youngster as his long-term successor.