The song "Love Me" was written by Jerry Leiber and Mike Stoller. Conceived as a parody of country and western music, it was initially recorded by R&B duo Willy & Ruth in 1954 (Spark 105), garnering a review spotlight in Billboard on August 14. Willie Headen was the lead singer of a vocal group, the Honey Bears and Ruth was the wife of another group member.
The following clip shows more artists that recorded the song, while the sound is from the Willy & Ruth original recording:
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The original label:

Spark Records was a record label started by Jerry Leiber and Mike Stoller in January 1954. Artists released on Spark Records included Willy & Ruth, The Sly Fox, Ervin "Big Boy" Groves, and The Robins. Leiber and Stoller eventually decided that while they wanted to write songs and make records, but did not want to deal with the mechanics of publishing records, so they sold the label to Atlantic Records.
For a good overview of songs recorded at Sparks Records, this Ace Records production is highly recommended:

http://www.acerecords.co.uk/the-spark-records-story
Shortly after its release, The Billy Williams Quartet released the song as well. In September 1954 on Coral 61264. In this quartet: Billy Williams, John Bell, Claude Riddick Jr. & Eugene Dixon:
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Billy Williams (December 28, 1910 – October 17, 1972) was an African-American singer. He had a successful cover recording of Fats Waller's "I'm Gonna Sit Right Down And Write Myself A Letter" in 1957. The record sold over one million copies, and was awarded a gold disc. His trademark hook for his songs was to shout "Oh, Yeah" at the end of lyrics. On April 5, 1959, the Billy Williams Quartet appeared on the Ed Sullivan show and performed a rousing version of Goodnight Irene.
Released in October 1954. A version by The Five DeMarco Sisters on BRUNSWICK 45-05349:
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The DeMarco Sisters were an American close harmony singing group of the big-band era who recorded popular music and performed in concerts and on the radio, television, and on film from the 1930s through the 1960s. They first achieved fame as weekly performers on The Fred Allen Show from 1946 to 1949, and were featured singers in the 1952 film Skirts Ahoy! with actress Esther Williams. The group was initially composed of five biological sisters.
Released in October 1954 as well. An early version of Love Me by Connie Russell on Capitol F2933:
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Connie Russell (May 9, 1923 – December 18, 1990) was an American singer and movie actress. Born in New York City, she appeared in seven films from the 1930s through the 1950s. She was far better known as a singer than as an actress, as her singing career was quite extensive.
It might be well possible that Elvis heard the song as sung by Jimmie Rodgers Snow. He released his version in January 1955 on RCA Victor 5986:
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Jimmie and Elvis on December 21, 1957:

Elvis recorded the song on September 1, 1956...
... For his second album, Elvis (RCA Victor, LPM-1382), issued on October 19.


It was also released on the EP, Elvis Vol. 1 (RCA Victor, EPA-992).

It climbed to the #2 position on the Billboard Top 100 in the United States, a first for a title not coming from single. "Love Me" also peaked at number seven on the R&B chart. "Love Me" was not released as a single to avoid confusion with Presley’s "Love Me Tender".
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Elvis sang "Love Me" on the October 28, 1956, Ed Sullivan Show as well.
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He included this song in the 1968 Comeback Special:
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And he often performed it in concerts in the seventies, including his last tour in June 1977.
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A few other versions:
Johnny Burnette, as released on July 1960 on his album "Dreamin'"
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The Polish singer Bogusław Wyrobek in 1960. For those who think Elvis had no influence in Eastern Europe

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Department Of Eagles, 2009:
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Sources
https://en.wikipedia.org/wiki/Love_Me_(Leiber/Stoller_song)
https://secondhandsongs.com/work/31525/versions#nav-entity
http://www.vocalgroupharmony.com/love_me.htm
https://en.wikipedia.org/wiki/Spark_Records
https://en.wikipedia.org/wiki/The_DeMarco_Sisters
https://en.wikipedia.org/wiki/Billy_Williams_(singer)
https://en.wikipedia.org/wiki/Connie_Russell
https://www.megocollector.com/elvis-presley/rca-lpm-1382-elvis-elvis-presley-undocumented-variation-confirmed/