Marr ignored the offer, Morrissey wrote before referring to himself in the third person.
Morrissey undertakes a largely sold out tour of the USA in November.
Marr continues to tour as a special guest to New Order.
Johnny Marr (left) and Morrissey of the group The Smiths posed with their arms around a tree trunk in London in 1983.Clare Muller/Redferns/Getty Images
In a new public statement, Marr says his former bandmate didnt accurately convey what happened.
I didnt ignore the offer, he wrote.
It prohibits Morrissey from using the name whilst also denying Morrissey considerable financial livelihood.
Morrissey alone created the musical unit name The Smiths in May 1982.
Marrs management team says this also isnt totally accruate.
In the aftermath, Marr reached out to Morrisseys team in an effort to jointly secure the bands name.
This I have done on behalf of both myself and my former bandmates.
In 2008, however, they did meet at a pub to discuss a possible Smiths reunion.
For four days it was a very real prospect.
It felt apart after Marr went to Mexico to tour with the Cribs.
And then suddenly there was radio silence, Marr wrote.
