Hawksworth said before Wednesday's game that no decision has been made on whether to place him on the DL.
The Dodgers would not confirm any roster move.
Hawksworth would be the Dodgers' 12th disabling injury this season, a number they didn't reach last season until June 16. In 14 appearances, Hawksworth is 1-1 with a 3.12 ERA while demonstrating a fastball that has been clocked as high as 98 mph.
Elbert, a first-round pick in the 2004 First-Year Player Draft, has 13 appearances for Albuquerque with a 2-0 mark and 5.02 ERA, apparently still plagued by wildness. In 14 1/3 innings, Elbert has issued nine walks with 16 strikeouts. Seven of his eight earned runs allowed have come in two appearances.
Ken Gurnick is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.