Under ASME, ASTM materials are NOT automatically accepted.

SA designated material should be used at all time ( NOT ASTM A designation), with following exception from Interpretation VIII-1-92-74:

Hence, the inspector MUST verify that the Year edition of the ASTM specification detailed on material certification is accepted by the specified edition of ASME code section II, Part A.

ASME II Part A details which  ASTM editions for each material are equivalent to ASME SA grade as below extract:

Example:

ASTM A-36 (2005) is identical to ASME SA-36 (ed. 2007/2008 add)

Meanwhile,  the ASTM A737 material is confirmed to be compliant to ASME SA if it certified to ASTM A737 from 1987-1999, so it is suitable for ASME VIII division 1 BUT NOT ASME VIII Division 3.