Targett joined Boro on loan in the summer on a season-long loan, but the deal included a clause that would allow Newcastle to recall the full-back this month.
The deadline for that clause was initially set for earlier this month but was extended to the end of the window by the Magpies as they battled an injury crisis that left them with just four defenders fit at one stage in January.
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The picture is now more positive for Newcastle, who have welcomed Dan Burn back into the side in the last week. Burn started the midweek Champions League draw at PSG and was again in the starting XI for Saturday night's 4-1 defeat at Liverpool.
But Sven Botman was forced to sit out the Liverpool defeat with tightness in his thigh, and Tino Livramento and Emil Krafth are still sidelined through injury.
Targett has established himself as a key figure at Boro this season and is a regular in Kim Hellberg's side.
The 30-year-old is understood to want to see out the season at Boro, where he has impressed and is effectively playing for his future. Targett is due to become a free agent in the summer when his contract at Newcastle expires.
Hellberg has spoken openly and repeatedly about his hope that Targett sees out the season at the Riverside, but the decision is in Newcastle's hands.
And asked for an update on Saturday night, Howe said: "That's something we'll be discussing until the end of the window."
Boro have signed a left-back, with Fin Munroe joining from Aston Villa earlier this month. But Munroe is just 20 and doesn't have any previous experience of playing in the Championship.
Quizzed on Targett last week, Hellberg said: "He's happy here. I think he’s an unbelievable character, such a good person with everyone, always helping everyone, always happy. He’s top.”