There... might just be a way. But it is going away far from what is known to be a Turing Machine.
As PJTraill wrote, we have to get some idea of "infinite steps". I propose our special machine $S$ has (countably) infinite tapes. Given a TM $T$ and input $x$ our infinite steps $S$ would do in finite steps would be that on tape $i$ the content of the tape of $T$ and its state on input $x$ after $i$ simulated steps of $T$ are encoded. We further should allow $S$ to have at least countably infinite many states. After each tape $i$ has been finished being constructed (which, as we remember basically happens in the traditional TM way), $S$ assumes a special tape state $0.a_1\ldots a_i$ with $a_i=1$ if the tape $i$ does not differ from tape $i-1$ and else $0$. When we now allow some mechanism that, after doing all the infinite steps in finite time, changes from our always finitely digited $0.a_1\ldots a_i$ to some infinite sequence $0.a_1\ldots$ - or in more mathematical terms, allow our machine to calculate the limit of the (in general uncomputationable) sequence of special tape stape states $q_i$ when $i\rightarrow\infty$ - then we could be finished, when the state $0.0000\ldots$ returns "TM halts" and all others with infinite digits return "TM does not hold".
I would like to add two things. First, this machine $S$, while halting after a finite number of steps by definition, could not be used to prove that it or machines with the same generalizations would halt. We are using the fact that a TM only has a finite number of states, so it can be encoded on each tape of $S$ using only finite space. Only using finite space on a tape is important so the tape itself can be traversed in finite steps. $S$ is so close to a TM as possible and as long as no infinite steps are done by it, its evaluation to a certain fixed point can be simulated by a normal TM. We were also using that the simulated TM only has (one or) finitely many tapes. We can simulate any such TM by a 1-tape TM, but with countable finite tapes, I don't know but doubt that's possible.
The second thing to mention is that this machine $S$ is by far not the thing Turing and like-wise minds of the time envisioned. They thought about what would be possible for a computator, e.g. a human with instructions, to calculate. It does not seem like this constructed machine $S$ would fall into this category.