Helene Demuth is who the claim is talking about -- she joined the Marx family in 1845 after being sent by Marx's mother in law.
Helene had a son in 1951, which Engels claimed paternity of, and who was later given up for foster care (though Marx's daughters maintained correspondence with him). The theory, which has been around since 1898, is that Engels only claimed this to save Marx's marriage with Jenny.
The evidence for this is a letter written in 1898 by Louise Strasser (who was married to Karl Kautsky until 1889) claiming that Engels made a deathbed confession to Marx's daughter Eleanor in 1895. The claim goes that Engels was worried about his name being "besmirched" -- a questionable idea given Engels' history and the fact that this was a 48 year old non-issue at this point. She also claims to have previously seen a letter between Engels and Marx confirming this, but suggests that Engels probably destroyed it (another questionably idea if he was worried about his good name). No such letter has ever been found.
Coincidentally Louise's letter was only written in the months after Eleanor herself had died, so she couldn't confirm or deny it. There's no other evidence that she, Helene, Engels, Marx, Jenny (who was extremely close to Helene or anyone else ever independently mentioned it.
Louise's letter suffers from other problems, including obvious inaccuracies. For example, it (rather oddly) claims that Engels also 'confessed' on his deathbed that Marx and Jenny had stopped having sex by 1950, but Jenny gave birth several times after this. None of Louise's contemporaries, including her ex-husband Kautsky, considered her a reliable source of information and almost everyone at the time dismissed the letter.
It's also worth mentioning that Frederich, the son, also once claimed in a letter to be Marx's son. But he was doing so in order to claim the right to some money -- which he didn't get. He offered no further evidence for the statement, and as far as we know never mentioned it in correspondence with Marx's daughters.
So there is a basis for the claim -- they didn't just invent it out of thin air. However, it's thin at best, and even that thin layer has some serious credibility issues.
Analysis from: https://old.reddit.com/r/communism101/comments/5m55ir/is_there_any_evidence_that_karl_marx_impregnated/dc11bcw/