Air carrier Captain reported ATC advised them of a nearby primary radar target during departure climb. The flight crew reported observing Unidentified Aerial Phenomena (UAP) that appeared to maintain the same track as the aircraft.

Narrative

During climb out on departure from ZZZ-ZZZ1 on Aircraft X; we encountered an unidentified flying object or phenomena of some sort approximately between the MSLIN and STOMP fixes. The object was brought to our attention by ATC when they advised us of a primary radar target that was at our 9 o'clock position within a half-mile distance. The close proximity of the traffic advisory raised our attention to the threat. On the captain side about the 8 o'clock position I observed the aircraft at the same altitude (FL250; during an extended level off during climb). The aircraft appeared to maintain the same track and during a period of time tracked at a further distance to the 7 o'clock position where it was determined no longer a threat. The aircraft appeared small and dark colored; potentially a large drone; single ship military fighter/trainer aircraft; or as large as a small business-type jet aircraft. The aircraft must have been traveling at least 200 knots; but probably not more than 250 knots; given our speed overtake when traveling at 290 knots with an approximate 20 knot headwind. ATC was interested in the visual report since they were not talking with the aircraft; nor aware of its presence apart from the small primary radar return; and it was not transmitting an ADS-B OUT signal despite the Class A airspace requirements. This phenomena represents a significant threat to our operation given the potential for a mid-air collision without the ability to react to a TCAS II Resolution Advisory as trained for the given scenario. Coordinate reports with controlling authorities to reduce potential for mid-air collision.