The problem of univariate polynomial approximate greatest common divisors (GCD) is a fundamental problem in the field of Symbolic-Numeric hybrid computation, and it is also important in some application fields. Estimation of sparse approximate GCDs of two univariate polynomials is studied in the paper. Although the computation of univariate polynomial approximate GCDs has been studied extensively, there are only a few articles concerning the problem of estimation of sparse approximate GCDs. Two algorithms have been established in the paper. One is a co-factors based sparse optimization model (MVL1), \begin{eqnarray*} \widehat{\textbf{c}}=\arg\,\min_{\textbf{x}}({\lambda\norm{\textbf{x}}}_{1}+\frac{1}{2}{\norm{M\textbf{x}-V}}_{2}^{2}), \end{eqnarray*}where $M$ is a matrix constructed by co-factors, $V$ is a vector stacked by the coefficient vectors of the two given polynomials, and $\lambda$ is a trade-off parameter. This is essentially established based on Sylvester subresultant matrix's right null space, and it is a modification of an algorithm in the literature. Another is a Sylvester subresultant matrix's left null space based sparse optimization model (SubRes.SSL1), \begin{equation}{\label{formul000} \widehat{\textbf{c}}=\arg\,\min_{\textbf{x}}(\lambda\norm{\textbf{x}}_{1}+\frac{1}{2}{\norm{Q\textbf{x}}_{2}^{2}})\,\,\,\,\,\,s.t.\,\,\,\,\,\,\textbf{v}^{T}\textbf{x}=1 }, \end{equation} where $Q$ is a matrix constructed via left null vectors of the Sylvester subresultant matrix, $v$ is the smallest singular vector of $Q$, and $\lambda$ is a trade-off parameter. To establish this algorithm, we have firstly established a subspace algorithm (SubRes.SS) to calculate approximate GCDs of two univariate polynomials. As stated in the algorithm, $v$ is actually a coefficient vector of an approximate GCD. We have compared our subspace method (SubRes.SS) with a subspace algorithm which is derived from a generalized Sylvester matrix existing in the literature. Numerical experiments show that our algorithm has better efficiency. Furthermore, we give numerical experiments to demonstrate estimation error of algorithms MVL1 and SubRes.SSL1 for estimation of sparse approximate GCDs. In each test, we construct two univariate polynomials and they have a GCD with sparse coefficients, specifically, the GCD coefficient vector has nonzero elements only in some specified positions. Then we add noise to the two polynomials. And then we estimate a sparse approximate GCD via the two perturbed polynomials by applying the algorithms MVL1 and SubRes.SSL1, respectively. Numerical experiments show that SubRes.SSL1 gets better estimation for the sparsity of GCD in general, but it gets similar estimation error for the nonzero coeffcients of GCD compared with MVL1.