From an interview with the composer of the song, Burt Bacharach (born: May 12, 1928) in May 2015:
Source: https://www.theguardian.com/music/2015/may/21/30-minutes-with-burt-bacharach-interviewAny great singers you wish you’d have worked with? Could you have worked with Elvis Presley, for example?
I doubt it, although he did record Any Day Now, and that’s one of those things where you think, “Great!”, and you hear it and it’s not so great.
He also composed the music for "I'll Never Fall In Love Again" with Hal David.

Bob Hilliard (January 28, 1918 – February 1, 1971) wrote the words for Any Day Now.
The original version of the song was recorded by Chuck Jackson in February 1962 and reached number 23 in the Billboard Hot 100 chart and number 2 in the R&B chart.
Here we see him in 1965 with composer Bacharach on the organ:
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Elvis recorded his version on February 21, 1969 at the American Sound Studios. It became the B-side for his hit-single "In The Ghetto" which was released in April that year:
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Some Early versions - released before February 1969
Dee Dee Sharp - 1962
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Peter and Gordon - July 1965
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Tom Jones - January 1967
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Oscar Toney Jr - 1967
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Some Later versions - released after April 1969
The composer himself recorded Any Day Now, and released it in May 1969. That’s one of those things where you think, “Great!”, and you hear it and it’s not so great:
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Faron Young - 1982
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Ronnie Milsap - 1982
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Steve Tyrell - February 2015
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Sources
http://www.poplartunes.nl/any_day_now.html
https://www.youtube.com/watch?v=9_OS04Sn1Oo
https://secondhandsongs.com/work/5995/versions#nav-entity