Light aircraft Flight Instructor reported an NMAC with a helicopter in the vicinity of BMI airport.

Narrative

On VFR approach into BMI; we heard helicopter traffic reporting 13 miles southeast as we were 13 miles to the southwest. As we entered a left downwind for Runway 29; the helicopter reported 7 miles to the southeast. Our Traffic Alerting System showed a target 300 ft. below our altitude approximately on the extended left downwind; opposite direction. We took evasive action to avoid the traffic once we spotted them. At this point; the traffic was 2 miles southeast of the airport when they passed off of our wing; about 200 ft. low and 300 ft. to our right. The problem was caused by inaccurate situational awareness resulting from incorrect position reports from the helicopter. Based on their distance reports; we expected them to be outside the Delta [airspace]; and it was clear that Tower thought the same. Due to the mismatch between the reports we were hearing and the traffic position we saw on our traffic map; we didn't take action to get out of the way soon enough. Unsure whether the helicopter's report over the radio or our traffic map was mistaken; we didn't take action to get out of the flight path of the helicopter until we had them in sight; which was a little too late. I don't believe Tower had any way to prevent this from happening today; though some kind of radar for them might prevent this kind of incident in the future. Though we were never explicitly given a traffic report; we also should have spoken up and asked about the traffic. Additionally; after we reported the incident to Tower; there was no follow-up. I don't know if they ever realized the mistakes that occurred; but I would have liked to get a chance to discuss what happened with them.