Read Spycatcher by Peter Wright, incredibly interesting story of a working class MI5 scientific intelligence officer (engineer) turned literal spycatcher - largely spending his time chasing the work of naive upperclass communists of years gone by.
During Hollis's tenure, the KGB and GRU were just one step ahead every single move they made, to the point that they broke up a spy ring and the KGB didn't even phone home to Moscow that day. Other examples include bugged rooms suddenly being vacated or bugs in walls being filled with needle-like precision. They gradually worked up the tree until they realized there had to be a mole very high up in MI5 (or not at all, but it's not as neat).
British intelligence had been completely rocked by the Cambridge Five - communists had infiltrated pretty much everywhere in British intelligence until people like Peter Wright modernized the service largely by running around chasing leads from 40 years ago (in the 1970s) from people like Anthony Blunt who had given up fighting. The British establishment had to be dragged kicking and screaming towards the notion that an Englishman could be a communist despite his upbringing and public school training.
Can you imagine the fuss then, when it turns out that Hollis - who is already a very conservative (unambitious, won't step on toes etc.) leader of the service - is suspected of treachery? He had spent years in China in his youth, possibly having contact with known Russian assets. If he was really a spy, the smoking gun is that it turned out that when he was sent to interview Igor Gouzenko (A defector now residing in Canada), he apparently showed almost no interest in what Gouzenko had to say and was wearing a disguise. This may sound relatively innocuous but remember that a defector could quite possibly recognize you on from photographs of their agents.
The main case against this hypothesis relies on a few fairly not overly strong but nonetheless useful points: Oleg Gordievsky (Briefly Deputy Head of the KGB in London) clearly stated that his station chief stated that this rumour around Hollis must've been some British trick. This is potentially very strong evidence, however, it is the word of one man. Gordievsky is reliable, however, but more importantly he was KGB and Hollis was always supposed to be GRU (and he was definitely inactive long before Gordievsky would've been able to speak English let alone stationed in London). ELLI is a currently unknown cryptonym (alleged to be Hollis), it may apparently be Leo Long but that doesn't add up with the leaks from MI5.
MI5 (via Andrew) and MI6 (via Macintyre's book) - both had official archive access, extremely rare for MI6 - both claim its smoke without fire, Misguided conspiracy. It could well be, but this glosses over that there was a (according to Wright at least) an investigation that lasted the best part of a decade neither indicted nor exonerated him (https://www.margaretthatcher.org/document/104603) but on balance said it was more likely that not that he was not a spy. However, the proceedings of said investigation are not public as far as I can and the investigation-investigation thatcher speaks of in the speech linked to was run by a friend of Hollis. Chapman Pincher seems to allege that Hollis's family spoke openly of his work for the USSR but Pincher is now dead, and I don't really believe him at face value.
https://www.iwp.edu/wp-content/uploads/2015/04/20150417_Repo... This document lays out pretty much everything known about Hollis, along with some (IIRC) elsewhere uncommon information linking Hollis's movements with that of the spy SONIA.
The real story here isn't really whether Hollis was a spy or not (he was incompetent as far as I can see), but that the Russian's are unbelievably good at espionage (but not unbeatable - see Operation PIMLICO).