Let $(X_{n, j}, j=1, 2, \\ldots; n=1, 2, \\ldots)$ be a row-wise triangular array of independent geometric random variables with probabilities\n$P\\left(X_{n, j} = k \\right) = p_{n, j}\\left(1-p_{n, j} \\right)^{k}, p_{n, j}\\in (0,1); j=1, 2, \\ldots; n=1, 2, \\ldots ; k=0, 1, \\ldots.$ Write $S_{n}=\\sum\\limits_{j = 1}^{n} X_{n, j}, \\lambda_{n} =\\sum\\limits_{j = 1}^{n} \\left(1-p_{n, j} \\right) p^{-1}_{n, j}$ and $\\mu_{n} =\\sum\\limits_{j = 1}^{n} \\left(1-p_{n, j} \\right).$ Denote by $Z_{\\lambda_{n}}$ and $Z_{\\mu_{n}}$ the Poisson random variables with parameters $\\lambda_{n}$ and $\\mu_{n},$ respectively. Let $\\mathbb{K}$ denote the class of all real-valued bounded functions on the set of all non-negative integers $Z_{+}=\\{0, 1, 2, \\ldots, n\\}.$ The norm of function $f\\in \\mathbb{K}$ is defined by $\\left\\| f \\right\\|\\mathop{= \\sup}\\limits_{x\\in Z_{+}}\\left| {f\\left( x \\right)} \\right|.$ The Renyi\'s operator associated with random variable X, denoted by $A_{X},$ is given by\n$$\nA_{X}f(x) = E\\big(f(X+x)\\big)=\\sum\\limits_{k = 0}^{\\infty} f(x + k)P(X = k), \\forall f \\in \\mathbb{K}, \\forall x \\in Z_{+}.\n$$\nLet $A_{S_{n}}, A_{Z_{\\lambda_{n}}}$ and $A_{Z_{\\mu_{n}}}$ denote the Renyi\'s operators associated with $S_{n}, Z_{\\lambda_{n}}$ and $Z_{\\mu_{n}},$ respectively. The main purpose of this paper is to establish the upper bounds for $\\parallel A_{S_{n}}f-A_{Z_{\\lambda_{n}}}f \\parallel$ and $\\parallel A_{S_{n}}f-A_{Z_{\\mu_{n}}}f \\parallel$ in Poisson approximation for independent geometric random variables. Some bounds related to $\\parallel A_{S_{N_{}}}f-A_{Z_{\\lambda_{N_{n}}}}f \\parallel$ and $\\parallel A_{S_{N_{n}}}f-A_{Z_{\\mu_{N_{n}}}}f \\parallel$ in Poisson approximation for random sums of independent geometric random variables are also investigated, with $N_{n}, n=1,2, \\ldots$ are positive integer-valued random variables independent of all $X_{n, 1}, X_{n, 2}, \\ldots; n=1, 2, \\ldots.$ The received results in this paper are extension and generalization of known earlier ones.