Shares in both companies surged in premarket trading Friday after Fast Company reported the two were discussing a merger deal. It cited several people with knowledge of the matter.
The report said the two companies first started discussing a partnership last summer but decided "a merger is the best path forward."
Target shares were up fractionally in Friday after jumping during the premarket when the report was published. Kroger's stock rose 8 percent initially on the report, but later pared much their gains.
The two have had meetings over a Shipt partnership, CNBC's source said. Target acquired same-day delivery company Shipt for $550 million in cash in December, in a move to bolster its supply chain and compete with rivals on speed.