In the United Kingdom, the song was a modest hit when released in 1992, but following a reissue in 1993 in the wake of its American success, it reached a new peak of number 15 on the UK Singles Chart.

[5] John T. Davis from Austin American Statesman complimented "Constant Craving" as "a song graced by a compelling melody and a vocal brimming with yearning.

Lang's distinctive voice is woven into spiraling multitracked harmonies, wafting over a base of acoustic guitar strumming and subtle accordian [sic] and xylophone interludes.

"[7] Steve Morse from Boston Globe named it the best song of the album, declaring it as "a romantic number that's also the only rhythmic tune of the bunch.

"[13] John Milward from Rolling Stone felt the song is "more elusive", "but there's no denying the lift of its propulsive rhythms and joyful harmonies.

"[14] David Bauder from The Salt Lake Tribune praised "Constant Craving" as one of lang's best songs, remarking that it "concludes that a sense of longing is an inevitable part of life.

[17] In 2008, the Daily Vault's Jason Warburg viewed it as a "dynamite single", that "manages to meld the spiritual with the physical while conveying a feverish desire for both kinds of fulfillment."