The Washington metropolitan area has always been considered a top contender.
The newspaper reported the talks included "how quickly the company would move employees there, which buildings it would occupy and how an announcement about the move would be made to the public".
Two people close to the process said that if Crystal City was selected, Amazon was likely to move an initial group of several hundred employees into 1851 South Bell Street or 1770 Crystal Drive.
Amazon announced previous year that the retail giant was looking to open a second headquarters somewhere in the US, bringing with it as many as 50,000 new jobs.
However, the Post cautions that it does not mean the deal is certain and that other cities might have "similar discussions".
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British officials are keen to avoid the negotiations dragging on, with Brexit day scheduled for March 29. The Government source added: "People shouldn't get ahead of themselves".
The Washington Post reported Saturday that the e-commerce giant is in "advanced talks" with officials and developers in Crystal City, Va., near Reagan International Airport just outside Washington, D.C., about locating its sought-after second headquarters there.
In earlier tweets, Grella criticized another story in the Post - owned by Amazon founder and CEO Jeff Bezos - for speculating about the possible second headquarters location based on flight patterns of Bezos' jet. A decision is scheduled by the end of the year, with 20 cities in the USA and Canada in the running. You're not doing Crystal City, VA any favors.
A spokesperson for Amazon declined to comment.
Crystal City seems to the front-runner in the drawn-out competition to win Amazon's $5 billion second headquarters, widely known as HQ2.
He also hinted that Amazon compelled finalists to sign a nondisclosure agreement (NDA) that the leakers may be violating.