Old floor tiles likely to contain asbestos?

One of the first things I did when moving into my current house 2 or 3
years ago was to remove the carpet (!) from the kitchen. Underneath
were grey 9" square tiles, a bit dingy but better than a grotty
carpet.

However the tiles proved to be pretty weak and after a while started
to crack. The last straw was a washing machine leak which dissolved
the adhesive and loosened some tiles, which then basically began to
fall to pieces - the old tiles are rigid but very brittle and it got
to the point where I just had to remove the loose ones, revealing the
cement floor and black grout underneath.

I am planning to refit the kitchen but won't be able to afford a
proper job for another year or two, so as a cheap(ish) temporary(ish)
fix I spent the weekend removing all the remaining tiles and laying
self-adhesive vinyl tiles. So far so good.

However, some helpful person has pointed out that the old tiles more
than likely contain asbestos. Is this likely to be true? The house was
built in about 1964. There was quite a lot of accumulated dust under
the old tiles which I swept, vacuumed and mopped up before laying the
new tiles - is this hazardous stuff?

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If they're brittle they're probably 'thermoplastic tiles' and quite
harmless except for sharp edges when they break.