Air Carrier Captain reported a burning oil odor during takeoff and initial climb which prompted the flight crew to request an immediate return to departure airport. After a safe landing the flight continued to the gate where maintenance met the aircraft.

Narrative

Overweight Landing. Normal taxi; We departed Runway XX ZZZ. As TO power applied we began to notice burning oil smell on flight deck. I thought about rejecting but it was not my leg and this is not an uncommon thing and dissipates once engines are at pores for a moment. Continued TO and after rotation fumes continued and seemed to increase. Out of 1500 I told ZZZ Tower Controller we were having a fume event and needed an immediate return. I did not declare emergency since there was no visible smoke. We were handed to ZZZ TRACON Departure; turned at 3000 ft. Downwind for Runway XY. I dinged Cabin and asked if they smelled anything; the response was no. I told them we would be returning to ZZZ and they had 10 minutes; let me know if they see anything. During this time odor continued. Didn't get any better. We accomplished after After TO ck; cruz; descent ck and a quick set up and brief of ILS Runway XY. weather was CAVOK. We were over landing weight and deployed Gear and flaps in downwind to burn fuel. I did a rough and dirty calc and figured we would be at landing weight at touchdown. I sent acars to Dispatch and called gate freq in ZZZ to tell them we would need a gate right away. FO (First Officer) landed perfectly smooth touch. I noted we had touched at 450# over MAX landing weight. We taxied to fast to gate. I started APU for electric and shut down the Left engine and the result was increase in fumes. We pulled into gate and shut down RH engine and went on APU air. The fumes subsided and I suspect it was the RH engine dumping oil into the bleed air to the pack. Maintenance arrived and Mechanic said he smelled the oil as he entered the plane. Post flight I spoke with ALL the flight attendants and the rear positions said they had strong fumes. The #1 on call said there were no fumes. Some people simply have bad smell? IDK. A cursery review of the aircraft past history indicated high oil consumption of the RH engine. I had noted this and it was included in my preflight threat assessment briefing to the FO. I did not declare an emergency as the traffic at the airport was not heavy and my priority was getting on the ground ASAP. It was extremely high workload and I commend the FO for the teamwork. I think we covered the bases and actually had to extend the downwind an extra minute to get finished. The Tower was aware of the situation and I assume an alert 1 was issued. I felt further distraction of declaration was not helpful. I fully blame Company maintenance procedures for not identifying and repairing the high oil consumption on the RH engine. They dispatched an aircraft with an ongoing defect. Jets are not Piston engines and high oil consumption should not be deferred and dispatched.Tricresyl phosphate when burned and introduced into cabin can kill. The Airline has a fume event reporting policy and has taken adequate steps to report the downline event reporting. I feel Maintenance diminishes the serious nature of fumes in the cabin. This aircraft should have not been dispatched with high oil burn.