I have not found a source.
I would like to suggest that from the subject matter of the zemiros some could possibly be assigned to Friday night as follows.
“Menucho vesimcho” mentions the sixth day, “ki leshisho kol beruim veomdim”.
“Mah Yedidus” speaks about the evening “mai’erev mazminim”.
“Mah Yofis” cites being ready before Shabbos “nechonim mib’oid yom”.
“Yom Shabbos Kodesh Hu” mentions the two angels that accompany us home from shul Friday night “umalach kail ya’ane boruch”
and “Yom Ze LeYisroel” talks about the completion of creation on the sixth day “beshaishes kiliso meleches olomim”.
“Tzomo Nafshi” seems to be a preface to “Nishmas” and might be assigned to Shabbos day.
“Koh Ribon Olam” seems not to be assignable.
“Koh ecsof noam” has a phrase about waiting for Shabbos kodesh to arrive “shomrei umetzapim Shabbos kodesh” and so can be justified for Friday night.
I could find not reason to assign the Shabbos day zemiros, except perhaps the mention of the “Lechem Haponim” (which were divided between the kohanim on Shabbos day) in “Ki Eshmero”.