The idea of a constitution certainly existed!
The notion of compiling a codified body of laws was around at least since 7th century BC. Laws of Draco and Solon in Athens (which we do not have), laws of Gortyn (which we do have as inscription). Rome herself had the Twelve Tables (whose content is unknown but whose existence is abundantly attested).
Incidentally, it was in 2nd century AD that Twelve Tables were formally repealed.
Now, as for constitution - a text which purports to give a comprehensive overview of the composition and functions of the higher organs of government, but omits private law - several existed. For example, we have constitution of Erythrai (5th century BC) and constitution of Cyrene (3rd century BC).
Athens had a number of constitutional reforms after the Laws of Solon. But what is not clear is whether the laws of Solon were formally repealed and replaces by any single document. Rather it seems that the reforms of Cleisthenes and Ephialtes were separate acts that did not purport to be a single codified constitution, but leave previous laws in force insofar as not superseded.
Now, Rome had constitutional reforms. Laws of Licinius and Sextus, Lex Hortensia, reforms of Gracchi and of Sulla, and of Pompey and Crassus, laws of Caesar and Augustus.
But it seems that there was no consolidated text purporting to be the constitution of Rome, and issuing such may have been deliberately avoided by the reformers.
Consolidated constitutions were not generally unacceptable to Romans. There are foundation and reform charters of Roman colonies, such as Lex Ursonensis (we have 4 out of 9 tables).
Roughly who, in 2nd century, might undertake to issue a consolidated Constitution of Roman Empire?