Interim coach John Carver told the pair, who between them have spent 13 years with the Premier League team, that they were not going to be offered new contracts.
Both players, who are on coaching courses in Northern Ireland, said they understood the decision to let them go but were unhappy with the way the news was delivered.
"We were here (Northern Ireland) probably not even a day. We had just finished a lecture and John Carver rang me and told me the club were not going to offer me a new deal," utility player Taylor told Sky Sports.
"I had a chat and then he asked me to pass the phone to Jonas. It was unbelievable," added Taylor who has been with Newcastle since 2009.
Gutierrez, who has spent seven years at the club, returned to the first team this season after recovering from testicular cancer.
The Argentina defender, who scored a goal last month when Newcastle ensured their Premier League safety on the final day of the season, said: "I did not expect to be offered a new contract and it is not about that because you have to move on.
"I can understand the football issues and those type of things but, more important for me, is that they can pick up the phone and call me or my agent, and say ‘thank you but we are going to release you and it is going to be OK’. That is more the way."
(Reporting by Douglas Beattie, editing by Tony Jimenez)
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