Lawyer Mark McDonald said he was seeking permission from the Court of Appeal to take the "exceptional, but necessary, decision" to apply to reopen her case.
He claimed key medical expert Dr Dewi Evans had changed his mind over the mechanism of death involving three of Letby's victims.
Letby, 34, is serving 15 whole-life orders for murdering seven infants and attempting to murder seven others, with two attempts on one of her victims, between June 2015 and June 2016.
Following two trials at Manchester Crown Court, Letby also lost two bids to challenge her convictions at the Court of Appeal - in May and most recently in October.
Findings
Mr McDonald said: "The primary grounds of appeal at the previous hearings related to the admissibility before the jury of the evidence of the lead prosecution expert Dr Dewi Evans.
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