We develop several Euclidean numerical radius bounds for the product of two d-tuple operators using positivity criteria of a \(2\times 2\) block matrix whose entries are d-tuple operators. From these bounds, by using polar decomposition of operators, we obtain Euclidean numerical radius bounds for d-tuple operators. Among many other bounds, it is shown that \(\begin{aligned} w_e(\textbf{A})\le & {} \frac{1}{\sqrt{2}} \Vert \textbf{A}\Vert ^{1/2}\sqrt{\left\| \sum _{k=1}^{d} (|A_k|+|A_k^*|)\right\| }, \end{aligned}\) where \(w_e(\textbf{A})\) and \(\Vert \textbf{A}\Vert \) are the Euclidean numerical radius and the Euclidean operator norm, respectively, of a d-tuple operator \(\textbf{A}=(A_1,A_2, \ldots ,A_d).\) Further, we develop an upper bound for the Euclidean numerical radius of an \(n\times n\) operator matrix whose entries are d-tuple operators. In particular, it is proved that if \([\mathbf {A_{ij}}]_{n\times n}\) is an \(n\times n\) operator matrix then \(\begin{aligned} w_e([\mathbf {A_{ij}}]_{n\times n})\le w ([a_{ij}]_{n\times n}), \end{aligned}\) where each \(\mathbf {A_{ij}}\) is a d-tuple operator, \(1\le i,j\le n\) , \(a_{ij}=w_e(\mathbf {A_{ij}})\) if \(i=j\) , \(a_{ij}= \sqrt{w_e(|\mathbf {A_{ji}|}+|\mathbf {A_{ij}^*}|)w_e(|\mathbf {A_{ij}|}+|\mathbf {A_{ji}^*}|)}\) if \(i<j\) , and \(a_{ij}= 0\) if \(i>j\) . Some related applications are also discussed.
