He has a choice to either utter it with his tongue or recall it with his heart since some of the wordings of the hadeeth are possible to mean one or the other.
Al-Mula al-Qaari rahimahullah commented on the hadeeth by saying:
(أَنَّ هَذَا الْأَمْرَ) ، أَيْ: الَّذِي يُرِيدُهُ كَمَا فِي رِوَايَةٍ، وَيُسَمِّي حَاجَتَهُ أَوْ يُضْمِرُ فِي بَاطِنِهِ،
“[That this affair], meaning the one he intends as comes in a narration. And he names his need or recalls it in himself.” End translation. [Al-Mirqaatul Mafaateeh]
Ibn Hajar rahimahullah said:
وظاهر سياقه أن ينطق به ويحتمل أن يكتفي باستحضاره بقلبه عند الدعاء
“…What is apparent from the context is that he utters it [i.e. his need] and it is as well possible to mean he suffices himself by recalling it with his heart when making du’a…”. End translation. [Fathul Baari (1/171)]
Note: I have noticed a group of the shuyookh encouraging to utter it with one’s voice since it is understood from the meaning of expressing one’s need.
AbdulFattaah Bin Uthman