This paper presents studies of Bose-Einstein correlations (BEC) in proton-proton collisions at a centre-of-mass energy of 13 TeV, using data from the ATLAS detector at the CERN Large Hadron Collider. Data were collected in a special low-luminosity configuration with a minimum-bias trigger and a high-multiplicity track trigger, accumulating integrated luminosities of 151 $\mu$b$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ and 8.4 nb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ respectively. The BEC are measured for pairs of like-sign charged particles, each with $|\eta |$ < 2.5, for two kinematic ranges: the first with particle $pT$ > 100 MeV and the second with particle $pT$ > 500 MeV. The BEC parameters, characterizing the source radius and particle correlation strength, are investigated as functions of charged-particle multiplicity (up to 300) and average transverse momentum of the pair (up to 1.5 GeV). The double-differential dependence on charged-particle multiplicity and average transverse momentum of the pair is also studied. The BEC radius is found to be independent of the charged-particle multiplicity for high charged-particle multiplicity (above 100), confirming a previous observation at lower energy. This saturation occurs independent of the transverse momentum of the pair.