Commercial Fixed Wing
Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Situational Awareness
ATC Issue All Types; Conflict NMAC
General None Reported / Taken
Human Factors; Procedure
Being controlled by Lubbock Approach; we were being vectored for a visual approach. We called the airport in sight; but were not cleared due to a small aircraft 10 miles north of the field flying southeast. We then were about 7 NM NE of the airport flying a 260 heading perpendicular to final approach course. We were notified of the traffic; but never saw them throughout the entire encounter. We were cleared down to 5;000 feet and the small aircraft was to maintain VFR at or below 4;500. The small aircraft  called us in sight and Lubbock Approach told them to maintain visual separation. We were then cleared the visual approach for RWY 17R; and transferred to Tower frequency. After switching to Tower and flying approximately a 45 degree intercept to a 3 NM final; I noticed on TCAS that the small aircraft traffic looked to have turned to fly directly underneath us. At this point traffic was about 2-3 o' clock at 1-2 miles and approximately 500 feet below us. I began calling out the traffic's altitude in reference to ours to my First Officer (Pilot Flying). I also commanded that he leveled off our descent. When passing directly under us according to TCAS the traffic was only 200 feet below us. Then directly off our left only 100 feet below us. During all of this; we somehow DID NOT get a TA or an RA. We continued our visual approach and landed without any irregularities. After arriving at the gate; I called Approach to complain in which they replied he had us in sight and the only amount of separation that is required is just visual separation. Therefore; no regulations were broken. However; I felt unsafe and if a passenger had seen traffic passing below us with minimal separation; it does not reflect well on the company or 121 industry as a whole.
Air carrier Captain reported an NMAC with a small aircraft on approach to LBB airport.