Kafka was the Chiefs' quarterbacks coach/pass game coordinator the past two seasons. He has been an assistant under Andy Reid in Kansas City since 2017.
The NFL Network was first to report the news of Kafka's expected hiring.
The Giants had narrowed their offensive coordinator search to three finalists: Kafka, Houston Texans quarterbacks coach/pass game coordinator Pep Hamilton and Cleveland Browns wide receivers coach/pass game coordinator Chad O'Shea. All three interviewed with Daboll in New Jersey this week.
Daboll now has his coordinators set with Patrick Graham expected to run the defense, Thomas McGaughey the special teams and Kafka the offense. Graham and McGaughey were retained by Daboll from Joe Judge's staff. Judge was fired last month after two seasons.
Graham was a finalist for the Minnesota Vikings head-coaching job that is expected to go to Rams offensive coordinator Kevin O'Connell. Daboll said at his introductory news conference that Graham would return if he did not land a head coaching position in the current cycle.
McGaughey was offered the job to be the Carolina Panthers' special teams coordinator but turned it down. He also interviewed for similar positions with the Los Angeles Chargers and Chicago Bears before deciding to return to the Giants.
Kafka, 34, was drafted by the Philadelphia Eagles and Reid in the fourth round out of Northwestern in the 2010 NFL draft. He bounced around the league for the better part of six years before getting into coaching as an assistant at Northwestern in 2016.
The Chicago-born Kafka joined Reid in Kansas City the following year as the offensive quality control coach and helped mentor quarterback Patrick Mahomes as a rookie.
There was a belief for several years that Reid was especially high on Kafka and that he would make him the Chiefs' offensive coordinator should Eric Bieniemy get a head-coaching job. Bieniemy appears likely to get passed over again in this year's hiring cycle.