"Yes, we won't be retaining those two players. Their contracts came to an end," club coach Clinton Larsen told snl24.com/kickoff.
Shongwe, whose contract with Bidvest Wits expires on June 30, was on loan at Arrows last season, scoring twice in 21 games.
Sources have informed this website that he and Arrows could not agree on financial terms for a new deal.
Nonyane featured 17 times in all competitions in 2017/18 as he lost his place in the team to Zolani Nkombelo, who was signed at the start of this past campaign from Mbombela United.
Nonyane, who suffered groin and foot injuries in his two seasons at Abafana Bes'thende, joined the club from Mpumalanga Black Aces in 2016 along with Lerato Lamola, Wayde Jooste and Limbikani Mzava.
Arrows recently released striker Thomas Chideu, while fellow Zimbabwean attacker Kudakwashe Mahachi has been transferred to Orlando Pirates.
Larsen plans on replenishing his squad with four recruits, with Congolese hitman Mike Dombo currently on trial at the Durban-based outfit.
"We are definitely looking to bring on board two full-backs, a midfielder and a striker. So definitely we are looking to at least bring four players in," Larsen explained.
"The balance of the players will come from our MultiChoice Diski team."
Larsen's side will start the 2018/19 season against league champions Mamelodi Sundowns in the MTN8 quarter-finals in Pretoria in August .