Helicopter pilot inbound to BKL airport reported an NMAC with a jet on final approach to Runway 24R.

Narrative

Enroute from ZZZ to ZZZ1; BKL ATC was contacted at 10 miles east of the airport while over CGF westbound. Initial call stated our callsign; location (over CGF); altitude (2200MSL); and destination of ZZZ1. ATC replied with altimeter setting and 'transition approved'.I noted a minute or so later that an incoming jet was calling for the visual for 24R. ATC cleared the jet to land. I noticed traffic at 10 o'clock from our position and confirmed on TCAS the location; and that it was 1000ft above us. I called out the traffic to the crew so we all had 'eyes' on it. I did not think the traffic was the inbound jet because ATC hadn't alerted us or the jet of each other which is standard for BKL.We noticed the traffic altitude had dropped to 800 ft above us and was descending as well as closing distance.I told the flying pilot to begin a descent to maintain altitude separation. I thought the traffic might be the inbound jet and knew it would be crossing in front of us while descending from above us.We stayed focused on the jet as it approached our flightpath and altitude.The jet appeared to be within 200-300 ft laterally and vertically as it crossed from left to right of our flight path.If we had not descended from 2200 MSL to 2000 MSL; we would've likely had a midair collision. At no time were we alerted by ATC of the traffic and at no time did the jet acknowledge our presence.As soon as the jet was past us and clear; I called ATC and advised them that we had traffic in sight. Their response was 'are you coming into our airspace' which surprised us as we had just been approved for the transition 2 minutes before. I replied that we were entering the airspace and our current location at about 5 miles east of the airport. ATC again approved our transition and we continued to ZZZ1 without further incident.