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Katok conjectured that every $C^{2}$ diffeomorphism $f$ on a Riemannianmanifold has the intermediate entropy property, that is, for any constant $c\in[0, h_{top}(f))$, there exists an ergodic measure $\mu$ of $f$ satisfying$h_{\mu}(f)=c$. In this paper we consider a conditional intermediate metricentropy property and two conditional intermediate Birkhoff average propertiesfor flows. For a basic set $\Lambda$ of a flow $\Phi$ and two continuousfunction $g,$ $h$ on $\Lambda,$ we obtain$$\mathrm{Int}\left\{h_{\mu}(\Phi):\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{and }\int g d\mu=\alpha\right\}=\mathrm{Int}\left\{h_{\mu}(\Phi):\mu\in\mathcal{M}(\Phi,\Lambda) \text{ and }\int g d\mu=\alpha\right\},$$$$\mathrm{Int}\left\{\int g d\mu:\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{and }h_{\mu}(\Phi)=c\right\}=\mathrm{Int}\left\{\int g d\mu:\mu\in\mathcal{M}(\Phi,\Lambda) \text{ and }h_{\mu}(\Phi)=c\right\}$$ and$$\mathrm{Int}\left\{\int h d\mu:\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{and }\int g d\mu=\alpha\right\}=\mathrm{Int}\left\{\int h d\mu:\mu\in\mathcal{M}(\Phi,\Lambda) \text{ and }\int g d\mu=\alpha\right\}$$ for any$\alpha\in \left(\inf_{\mu\in \in \mathcal{M}(\Phi,\Lambda) }\int g d\mu, \,\sup_{\mu\in \in \mathcal{M}(\Phi,\Lambda) }\int g d\mu\right)$ and any $c\in(0,h_{top}(\Lambda)).$ In this process, we establish 'multi-horseshoe'entropy-dense property and use it to get the goal combined with conditionalvariational principles. We also obtain same result for singular hyperbolicattractors.

Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows