The sediments bearing this concrations are presumably terrigenious.
I know that concrations are diagenetic, but can we find connection between the water body type, that deposited the sediments and the presented concrations.
It is not travertine. It looks like chalk, but the problem is that chalk usually is with marine origin. Here the host sediments are presumably terrigenous.
From your photos (particularly the third photo), this looks like calcite accumulation (K horizon) in a calcareous soil. So, that would suggest a relatively arid or semi-arid climate. So, yes, "diagenetic" in the sense of formation of soil horizons, not diagenetic or depositional as in happening at the time of deposition or shortly after deposition. The concretions would form during a long period of soil development.
From all your photographs, they seems to caliche nodules as well as few rhizoliths; generally considered to be formed by meteoric diagenesis of the sediments in semiarid climate with perodic subaerial exposure.
I agree with Curtis, it seems to be pedogenetic nodular calcrete ror relict of them, I write relict because in the third pictures I think that the nodules where reworked from the original place either by colluvial processes and redeposited in the "stone line", or have undergone bioturbation and accumulated under this A horizon in a biogenic "stone line".
From the first and second pictures, your nodules seems to be in the host rock but the profile (outcrop) can have been eroded and the top is missing. The top should have been made from a more continous calcrete (calcic horizon in pedology). That is just a supposition, of course I need to be in the field for a better analyse.
I suggest you to look at papers on calcrete like this one:
Cheers
Article Development and lithogenesis of the palustrine and calcrete ...