In Genesis 24:2
Abraham said to his servant, the oldest of his household who was in charge of all that he owned, “Please place your hand under my thigh,
Who was this servant?
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We are not told the name of the servant here. Abraham had previous employed Eliezer of Damascus (Gen 15:2, 4) but whether this same man was still his chief steward is not known.
It is entirely possible that Eliezer of Damascus was still the chief steward of Abraham's household - he had been very well regarded as, in the absence of Ishmael and Isaac was destined to inherit all that Abraham had (Gen 15:2).
However, the fact that he is named in Gen 15 and remains un-named in Gen 24 suggests, but not disprove, that Eliezer had either died and been replaced, or that he was no longer employed by Abraham.
In Gen 15 Abraham is between 75 and 85 years old. In Gen 24 he is 140 years old. It is possible, but not essential that Eliezer had already died by simply being outlived by Abraham. However, if Eliezer were (say) 40 years younger that Abraham (??) then he might still be alive.
Benson believes that the servant in Gen 24 is still Eliezer as does the Jamieson-Fausset-Brown Bible Commentary. So does Gill, and the Pulpit commentary. They are possibly correct but we cannot be certain. Nothing in the story rules this out.