A person familiar with the discussions told The Associated Press that while talks "were heating up," but there was no "definitive" agreement. The person spoke on the condition of anonymity because the Sabres have declined to discuss their search to replace Ted Nolan, who was fired last month.
The person added that the interview took place in Buffalo and included Sabres owners Terry and Kim Pegula and general manager Tim Murray.
Buffalo's WKBW-TV posted a picture on its Twitter account of Bylsma, Murray and Sabres President Ted Black walking together in downtown Buffalo late Wednesday night.
Sabres spokesman Michael Gilbert said the team had nothing to report.
Bylsma spent five-plus seasons in Pittsburgh, where he won the 2009 Stanley Cup championship and was the NHL's 2011 coach of the year. He was fired a year ago after winning a franchise-best 252 games.
Though his contract doesn't expire until June 2016, he has the Penguins' permission to seek other jobs.
The Sabres and New Jersey Devils are the two remaining NHL teams with openings after the San Jose Sharks were prepared to hire Peter DeBoer.
Buffalo was forced to restart its search last week after losing out on hiring Mike Babcock. The former Red Wings coach turned down the Sabres' offer to instead sign with Toronto.
Bylsma and the Sabres have several mutual connections.
The Pegulas were previously based in Pittsburgh before purchasing the Sabres in February 2011.
Both have ties to USA Hockey. The Sabres have hosted numerous international tournaments and national events. Bylsma coached the U.S. National team to a fourth place finish at the 2014 Sochi Games, and was an assistant on the U.S. team that won the bronze medal at the World Championships two weeks ago.
Though the Sidney Crosby-led Penguins enjoyed plenty of regular-season success under Bylsma, they faltered in the playoffs since winning the Cup in 2009. They were twice knocked out in the first round, and reached the Eastern Conference final just once, in 2013, when they were swept in four games by Boston.
The Sabres are seeking their third coach since Lindy Ruff was fired in February 2013.
Buffalo is coming off consecutive last-place finishes and is in the midst of a top-to-bottom overhaul with the emphasis on building through youth. One key piece is expected to arrive in the NHL draft next month, when the Sabres are anticipated to use the No. 2 pick to select Boston University centre Jack Eichel.