The legislation, which Gov. Jack Markell has pledged to sign if passed by lawmakers, was scheduled for discussion Wednesday in the House Administration Committee.
The measure comes a little more than a year after Delaware first began recognizing same-sex civil unions.
Under the proposal, no new civil unions would be performed after July 1, and couples in civil unions could convert their civil unions to marriages before July 2014 by applying for a marriage license to county clerks of the peace.
On July 1, 2014, all remaining civil unions not subject to proceedings for dissolution, annulment or legal separation would automatically convert to marriages.