Jean-Camille Birget answered my question. These are called universally halting Turing machines. The oldest reference is:

Martin Davis (1956). A note on universal Turing machines. In Shannon, C. E., McCarthy, J., eds, Automata Studies, pp. 167-175. Princeton University Press.

Birget proved a complexity version of this: Every deterministic Turing machine with time complexity $T(n)$ is equivalent to a deterministic Turing machine which halts after $O(T(n))$ steps, no matter what configuration of size $n$ this machine starts in [J.C. Birget, Infinite String Rewrite Systems and Complexity, J. Symbolic Computation (1998) 25, 759-793.]

Jean-Camille Birget answered my question. These are called universally halting Turing machines. The oldest reference is:

Martin Davis (1956). A note on universal Turing machines. In Shannon, C. E., McCarthy, J., eds, Automata Studies, pp. 167-175. Princeton University Press.

Birget proved a complexity version of this: Every deterministic Turing machine with time complexity $T(n)$ is equivalent to a deterministic Turing machine which halts after $O(T(n))$ steps, no matter what configuration of size $n$ this machine starts in [J.C. Birget, Infinite String Rewrite Systems and Complexity, J. Symbolic Computation (1998) 25, 759-793.]

Update Friedrich Otto sent the following two more references:

Herman, G.T., Strong computability and variants of the uniform halting problem, Zeitschrift fuer mathematische Logik und Grundlagen der Mathematik, 17, 1971, 115--131

Shepherdson, J.C., Machine configuration and word problems of given degree of unsolvability, Zeitschrift fuer mathematische Logik und Grundlagen der Mathematik, 11, 1965, 149--175