I don't know what to add except that the creature is able to get ahead of him no matter what. The story is probably about 15-20 years old and in English. This is a big alien and I think the chase is by highway in the U. S.. I'm not sure why this happens, but by the end the man is desperate and scared.
Sounds a bit like Arthur Porges' The Ruum, except that it is a robot left by aliens which chases him.
The ruum is set to collect specimens weighing between 145lb and 175lb. The man weighs 149lb, but due to running away from it all through the night, by the time it catches him he has sweated off so many pounds that he has fallen below the lower weight limit, so that it just sniffs at him and goes away.