Let n be a non-zero integer. A set of m positive integers {a1, a2, · · · , am} such that aiaj + n is a perfect square for all 1 ≤ i <j ≤ m is called a Diophantine m-tuple with the property D(n). In a series of papers, Dujella studied the quantity Mn = sup{|S| : S has the property D(n)} and showed for |n| ≥ 400 that Mn ≤ 15.476 log |n| and if |n| >10100, then Mn <9.078 log |n|. We refine his argument to show that Cn ≤ 2 log |n| + O (log |n|/(log log |n|)2), where the implied constant is effectively computable and Cn = sup{|S ∩ [1, n2 ]| : S has the property D(n)}. Together with earlier work of Dujella, this implies Mn ≤ 2.6071 log |n| + O (log |n|/ (log log |n|)2), where the implied constant is effectively computable.