  We describe a recursive algorithm that produces an integral basis for the
centre of the Hecke algebra of type A consisting of linear combinations of
monomial symmetric polynomials of Jucys--Murphy elements. We also discuss the
existence of integral bases for the centre of the Hecke algebra that consist
solely of monomial symmetric polynomials of Jucys--Murphy elements. Finally,
for n=3, we show that only one such basis exists for the centre of the Hecke
algebra, by proving that there are exactly four bases for the centre of the
corresponding symmetric group algebra which consist solely of monomial
symmetric polynomials of Jucys--Murphy elements.
