The 32-year-old Argentine was released by the Magpies this summer after his contract expired.
Deportivo said the deal is subject to Gutierrez, who returned to action in March after recovering from testicular cancer, passing his medical.
Gutierrez scored 12 goals in 205 appearances for Newcastle after joining from Mallorca in 2008.
He criticised the Magpies' hierarchy after leaving St James' Park, saying they made him feel like they "didn't care" after releasing him in a phone call.
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Former Newcastle midfielder Jonas Gutierrez has agreed a move to Spanish top-flight side Deportivo La Coruna.